Skip to main content

Full text of "The Development Of Mathematics Second Edition"

See other formats

The Development of 





E. T. Bell 

Professor of Mathematics, California Institute of Technology 

Second Edition 


York • J^ondon 


Copyright, 1940, 1945, by 
E. T. Bell 


All rights reserved. This book, or 
farts thereof, may not be reproduced 
in any form without permission of 
the author. 



To Any Prospective Reader 

Nearly fifty years ago an American critic, reviewing the first 
volume (1888) of Lie’s Theorie der Transformationsgruppen, set 
his own pace (and ours) in the following remarks. 

There is probably no other science which presents such different appear- 
ances to one who cultivates it and one who does not, as mathematics. To [the 
noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, 
unambiguous reasoning. To the mathematician, on the other hand, his science 
is yet in the purple bloom of vigorous youth, everywhere stretching out 
after the "attainable but unattained,” and full of the excitement of nascent 
thoughts; its logic is beset with ambiguities, and its analytic processes, like 
Bunyan’s road, have a quagmire on one side and a deep ditch on the other, 
and branch off into innumerable by-paths that end in a wilderness.* 

Once we venture beyond the rudiments, we may agree that 
those who cultivate mathematics have more interesting things to 
say than those who merely venerate. Accordingly, we shall follow 
the cultivators in their explorations of a Bunyan’s road through 
the development of mathematics. If occasionally we have no 
eyes for the purple bloom, it trill be because we shall need all our 
faculties to avoid falling into the ditch or wandering off into a 
wilderness of trivialities that might be mistaken for mathematics 
or for its history. And we shall leave to antiquarians the difficult 
and delicate task of restoring the roses to the cheeks of mathe- 
matical mummies. 

The course chosen in the following chapters was determined 
by two factors. The first was the request from numerous corre- 
spondents, principally students and instructors, for a broad 
account of the general development of mathematics, with par- 

’ C. H. Chapman, Bulletin cf the Nets York Mathematical Society, 2, 1S92, 61. 



ticular reference to the main concepts and methods that have, 
in some measure, survived. The second was personal association 
for several years with creative mathematicians in both the pure 
and the applied divisions. 

Not a history of the traditional kind, but a narrative of the 
decisive epochs in the development of mathematics was wanted. 
A large majority asked for technical hints, where possible with- 
out too great detail, why certain things continue to interest 
mathematicians, technologists, and scientists, while others are 
ignored or dismissed as being no longer vital. Many who planned 
to end their mathematical education with the calculus, or even 
in some instances earlier, wished to be shown something of the 
general development of mathematics beyond that outstanding 
landmark of seventeenth-century thought, as part of a civilized 
education. Those intending to continue in mathematics or 
science or technology also asked for a broad general treatment 
with technical hints. They gave two additional reasons, the 
second of singular interest to any professed teacher. They be- 
lieved that a survey of the main directions along which living 
mathematics has developed would enable them to decide more 
intelligently in what particular field of mathematics, if any, they 
might find a lasting satisfaction. 

The second reason for their request was characteristic of a 
generation that has grown rather tired of being told what to 
think and whom to respect. These candid young critics of their 
would-be educators hoped that a cursory personal inspection of 
the land promised them, even from afar off, would enable them 
to resist the blandishments of persuasive ‘subdividers’ bent on 
selling their own tracts to the inexperienced. We seem to have 
come a long way since 1873, when that erudite English historian 
of mathematics and indefatigable manufacturer of drier-than- 
dust college textbooks, Isaac Todhunter (1820-1884), counseled 
a meek docility, sustained by an avid credulity, as the path of 
intellectual rectitude: 

If he [a student of mathematics] does not believe the statements of his 
tutor, probably [in Todhunter’s day at Cambridge] a clergyman of mature 
knowledge, recognized ability and blameless character — his suspicion is 
irrational, and manifests a want of the power of appreciating evidence, a want 
fatal to his success in that branch of science which he is supposed to be 

Be the wisdom of Todhunter’s admonition what it may, it is 
astonishing how few students entering serious work in mathe- 


matics or its applications have even the vaguest idea of the 
highways, the pitfalls, and the blind alleys ahead of them. 
Consequently, it is the easiest thing in the world for an enthusi- 
astic teacher, “of mature knowledge, recognized ability and 
blameless character,” to sell his misguided pupils a subject that 
has been dead for forty or a hundred years, under the sincere 
delusion that he is disciplining their minds. With only the 
briefest glimpse of what mathematics in this twentieth century 
— not in 2100 b.c. — is about, any student of normal intelligence 
should be able to distinguish between live teaching and dead 
mathematics. He will then be less likely than his confiding com- 
panion to drown in the ditch or perish in the wilderness. 

Many asked for some reference to the social implications of 
mathematics. A classic strategy in mathematics is the reduction 
of an unsolved problem to one already solved. It seems plausible 
that more than half the problem of mathematics and society 
is reducible to that of the physical sciences and society. There 
being as yet no widely accepted solution of the latter problem, 
we shall leave the former with the reduction indicated. Anyone 
will thus be able to reach his own conclusions from that solution 
of the scientific problem which he accepts. Proposed solutions 
range from Platonic realism at one extreme to Marxian deter- 
minism at the other. Occasional remarks may suggest an inquiry 
into the equally difficult question of w r hat part civilization, with 
its neuroses, its wars, and its national jealousies, has played in 
mathematics. These asides may be of interest to those intending 
to make mathematics their lifework. Incidentally, in this con- 
nection, I was told that I might write for adults. Chronological 
age is not necessarily a measure of adulthood; a first-year student 
in a university may be less infantile, in everything but mathe- 
matics, than the distinguished savant lecturing at him. 

The topics selected for description were chosen after con- 
sultation with numerous professionals who know from hard 
personal experience %vhat mathematical invention means. On 
their advice, only main trends of the past six thousand years are 
considered, and these are presented only through typical major 
episodes in each. As might be anticipated by any tvorker in 
mathematics, the conclusions reached by following such advice 
differ occasionally from those hallowed by the purely historical 
tradition. Wherever this is so, references to other accounts will 
enable any reader to form his own opinion. There are no abso- 
lutes (except possibly this) in mathematics or in its history. 



Most of the differences reflect two possible and sometimes 
divergent readings of mathematical evolution. Whoever has him- 
self attempted to advance mathematics is inclined to be more 
skeptical than the average spectator toward any alleged antici- 
pation of notable progress. From his own experience and that 
of others still living, the professional mathematician suspects 
that often what looks like an anticipation after the advance was 
made was not even aimed in the right direction. From many a 
current instance, he knows further that when at length progress 
started, it proceeded along lines totally different from those 
which, in retrospect, it ‘should 5 have followed. 

Nothing is easier, on the other hand, than to fit a deceptively 
smooth curve to the discontinuities of mathematical invention. 
Everything then appears as an orderly progression from the 
Egypt of 4000 b.c. and the Babylon of 2000 b.c. to the Gottingen 
of 1934 and the U.S.A. of 1945, with Cavalieri, for instance, 
indistinguishable from Newton in the neighborhood of the 
calculus, or Lagrange from Fourier in that of trigonometric 
series, or Bhaskara from Lagrange in the region of Fermat’s 
equation. Professional historians may sometimes be inclined to 
overemphasize the smoothness of the curve; professional mathe- 
maticians, mindful of the dominant part played in geometry by 
the singularities of curves, attend to the discontinuities. This is 
the origin of most differences of opinion between the majority 
of those who cultivate mathematics and the majority of those 
who do not. That such differences should exist is no disaster. 
Dissent is good for the souls of all concerned. 

No apology need be tendered the thousands of dead and 
living mathematicians whose names are not mentioned. Only a 
meaningless catalogue could have cited a tenth of those who have 
created mathematics. Nor, when between 4,000 and 5,000 
papers and books devoted to mathematical research — the cre- 
ation of new mathematics — are being published every 365 days, 
is there any point in attempting to minimize the omission of 
certain topics that have interested, and may still interest, 
hundreds of these unnamed thousands. However, what a suffi- 
cient number of competent men consider the vital things are at 
least mentioned. Anyone desirous of following the detailed his- 
tory of certain major developments will find the technical 
histories of special topics, written by mathematicians formathe- 
maticians, ample for a beginning. Some of these severely 
technical histories extend to hundreds of pages, a few to thou- 



sands; they refer to the labors of thousands of men, most of 
whom are all but completely forgotten. Yet, like the tiny crea- 
tures whose empty frames survive in massive coral reefs that can 
wreck a battleship, these hordes of all but anonymous mathe- 
maticians have left something in the structure of mathematics 
more durable than their own brief and commonplace lives. 

As to the mechanical features of the book, the inevitable foot- 
notes have been kept to a minimum by the simple expedient of 
throwing hundreds away. Some direct those seeking further 
information on the relevant mathematics to works by creative 
mathematicians. Other things being equal, preference is given 
works containing extensive bibliographies compiled by experts 
hairing firsthand knowledge of the subjects treated. A superscript 
number indicates a footnote; all are collected for easy reference just 
before the index. All should be ignored till a possible return to some 

The index will be found helpful. Men’s initials and dates 
(except a very few, unobtainable without undue labor), seldom 
repeated in the text, are given in the index; cross references to 
definitions, etc., are avoided by the same means. 

Nationalities arc stated; if more than one country has a 
claim to some man, the place w'here he did most of his work is 
given. On a previous occasion {Men of mathematics, Simon & 
Schuster, New York, 1937), I almost precipitated an inter- 
national incident by calling a Pole a Russian. I trust that few 
such disastrous blunders will be found here. The book mentioned 
contains full-length biographies of about thirty-five leading 
mathematicians of the past. 

Dates in the text appended to mathematical events serve 
two purposes, the first of which is obvious. The second is to 
avoid elaborate references. The date, if later than 1636 and 
earlier than 1868, will usually enable anyone seriously interested 
to locate the matter concerned in the collected works of the 
author cited; if later than 1867, and whether or not collected 
works arc available, the exact reference, with a concise abstract 
of the work, is given in the annual Jahrbuch ilbcr die Fortschrittc 
der Mathematik. For the period beginning in 1931, the Zcntralblatt 
fur Mathematik und Hire Grenzgebiete serves the same purpose. 
The American Mathematical Reviews, 1940-, is of the same 
general character as the German abstract journals. Compara- 
tively scarce early periodicals, likely tobc found only in specialized 
libraries, are not cited, although they were frequently con- 



suited. This omission may be partly compensated by refer- 
ring to the German and French mathematical encyclopedias 
listed in the notes. Other references to sources before 1637 are 
given in the proper places. 

For the period before 1637, the works of professional his- 
torians of mathematics have been used for some matters on 
which the historians arc in approximate agreement among them- 
selves. Theirs is a difficult and exacting pursuit; and if con- 
troversies over the trivia of mathematics, of but slight interest 
to either students or professionals, absorb a considerable part of 
their energies, the residue of apparently sound facts no doubt 
justifies the inordinate expense of obtaining it. Without the 
devoted labors of these scholars, mathematicians would know 
next to nothing, and perhaps care less, about the first faltering 
steps of their science. Indeed, an eminent French analyst of the 
twentieth century declared that neither he nor any but one or 
two of his fellow professionals had the slightest interest in the 
history of mathematics as conceived by historians. He amplified 
his statement by observing that the only history of mathematics 
that means anything to a mathematician is the thousands of 
technical papers cramming the journals devoted exclusively to 
mathematical research. These, he averred, are the true history 
of mathematics, and the only one either possible or profitable 
to write. Fortunately, I am not attempting to write a history 
of mathematics; I hope only to encourage some to go on, and 
decide for themselves whether the French analyst was right. 

Preference has been given in citing purely historical refer- 
ences to works in English, French, or German, as these are the 
three languages of which those interested in mathematics are 
most likely to have an adequate reading knowledge. For those 
especially interested in geometry, Italian also is necessary. 
Italian historical works are included in the bibliographical 
material of the histories listed. 

To the many professional friends who have advised me on 
their respective specialties and whose generous help I have 
attempted to pass on to others, I am very grateful. A special 
word of thanks is due Professor W. H. Gage, of the University 
of British Columbia, who removed many obscurities and greatly 
improved several of the presentations. 

This has been an opportunity to do something a little off 
the beaten track to show prospective readers how the mathe- 
matics familiar to them got where it is, and where it is going 



from there. I trust that students will tolerate the departure from 
the traditional textbook. For one thing, at any rate, the more 
sensible should be grateful: only the most ingenious instructor 
could set an examination on the book. 

It has, unhappily, been necessary in writing the book to 
consider many things besides the masterpieces of mathematics. 
Rising from a protracted and not always pleasant session with 
the works of bickering historians, scholarly pedants, and con- 
tentious mathematicians, often savagely contradicting or meanly 
disparaging one another, I pass on, for what it may be worth, 
the principal tiling I liave learned to appreciate as never before. 
It is contained in Buddha’s last injunction to his followers: 

Believe nothing on hearsay. Do not believe in traditions because 
they are old, or in anything on the mere authority of myself or any 
other teacher. 

E. T. Bell. 

Note to the Second Edition 

About fifty pages of new material have been added in this 
edition. The additions include numerous short amplifications of 
miscellaneous topics from Greek mathematics to mathematical 
logic, with longer notes on symbolism, algebraic and differential 
geometry, lattices, and other subjects in which there have been 
striking recent advances. 

E. T. Bell. 

California Institute of Technology, 

Pasadena, California, 

July, ms. 



To Any Prospective Reader v 


1. General Prospectus 3 

2. The Age of Empiricism 26 

3. Firmly Established 49 

4. The European Depression 85 

5. Detour through India, Arabia, and Spain 93 

6. Four Centuries of Transition, 1202-1603 107 

7. The Beginning of Modem Mathematics, 1637-1687 131 

8. Extensions of Number 167 

9. Toward Mathematical Structure 186 

10. Arithmetic Generalized 218 

11. Emergence of Structural Analysis 245 

12. Cardinal and Ordinal to 1902 270 

13. From Intuition to Absolute Rigor, 1700-1900 282 

14. Rational Arithmetic after Fermat 296 

15. Contributions from Geometry 320 

16. The Impulse from Science 361 

17. From Mechanics to Generalized Variables 370 

IS. From Applications to Abstractions 382 

19. Differential and Difference Equations 400 

20. Invariance 420 

21. Certain Major Theories of Functions 469 

22. Through Physics to General Analysis and Abstractness. . . 516 

23. Uncertainties and Probabilities 548 

Notes. 595 

Index 611 


The Development 
of Mathematics 


General Prospectus 

In all historic times all civilized peoples have striven toward 
mathematics. The prehistoric origins are as irrecoverable as those 
of language and art, and even the civilized beginnings can only 
be conjectured from the behavior of primitive peoples today. 
Whatever its source, mathematics has come down to the present 
by the two main streams of number and form. The first carried 
along arithmetic and algebra, the second, geometry. In the seven- 
teenth century these two united, forming the ever-broadening 
river of mathematical analysis. We shall look back in the 
following chapters on this great river of intellectual progress 
and, in the diminishing perspective of time, endeavor to see the 
more outstanding of those elements in the general advance from 
the past to the present which have endured. 

‘Form,’ it may be noted here to prevent a possible misappre- 
hension at the outset, has long been understood mathematically 
in a sense more general than that associated with the shapes of 
plane figures and solid bodies. The older, geometrical meaning is 
still pertinent. The newer refers to the structure of mathematical 
relations and theories. It developed, not from a study of spacial 
form as such, but from an analysis of the proofs occurring in 
geometry, algebra, and other divisions of mathematics. 

Awareness of number and spacial form is not an exclusively 
human privilege. Several of the higher animals exhibit a rudi- 
mentary sense of number, while -others approach genius in their 
mastery of form. Thus a certain cat made no objection when she 
was relieved of two of her six kittens, but was plainly distressed 
when she was deprived of three. She was relatively as advanced 
arithmetically as the savages of an Amazon tribe who can count 




up to two, but who confuse all greater numbers in a nebulous 

Again, the intellectual rats that find their way through the 
mazes devised by psychologists are passing difficult examinations 
in topology. At the human level, a classic puzzle which usually 
suffices to show the highly intelligent the limitations of their 
spacial intuition is that of constructing a surface with only one 
side and one boundary. 

Although human beings and the other animals thus meet on a 
common ground of mathematical sense, mathematics as it has 
been understood for at least twenty-five centuries is on a far 
higher plane of intelligence. 

Necessity for proof; emergence of mathematics 

Between the workable empiricism of the early land measurers 
who parceled out the fields of ancient Egypt and the geometry 
of the Greeks in the sixth century before Christ there is a great 
chasm. On the remoter side lies what preceded mathematics, on 
the nearer, mathematics; and the chasm is bridged by deductive 
reasoning applied consciously and deliberately to the practical 
inductions of daily life. Without the strictest deductive proof 
from admitted assumptions, explicitly stated as such, mathe- 
matics does not exist. This does not deny that intuition, experi- 
ment, induction, and plain guessing are important elements 
in mathematical invention. It merely states the criterion by 
which the final product of all the guessing, by whatever name it 
be dignified, is judged to be or not to be mathematics. Thus, for 
example, the useful rule, known to the ancient Babylonians, that 
the area of a rectangular field can be computed by ‘length times 
breadth,’ may agree with experience to the utmost refinement of 
physical measurement; but the rule is not a part of mathematics 
until it has been deduced from explicit assumptions. 

It may be significant to record that this sharp distinction 
between mathematics and other sciences began to blur slightly 
under the sudden impact of a greatly accelerated applied mathe- 
matics, so called, in the second world war. Semiempirical pro- 
cedures of calculation, certified by their pragmatic utility in 
war, were accorded full mathematical prestige. This relaxation 
of traditional demands brought the resulting techniques closer 
in both method and spirit to engineering and the physical 
sciences. It was acclaimed by some of its practitioners as a 
long-overdue democratization of the most aristocratic of the 



sciences. Others, of a more conservative persuasion, deplored 
the passing of the ideal of strict deduction, as a profitless con- 
fusion of a simple issue which had at last been clarified after 
several centuries of futile disputation. One fact, however, 
emerged from the difference of opinion: It is difficult, in modern 
warfare, to wreck, to maim, or to kill efficiently without a con- 
siderable expenditure of mathematics, much of which was 
designed originally for the development of those sciences and 
arts which create and conserve rather than destroy and waste. 

It is not known where or when the distinction between induc- 
tive inference — the summation of raw experience — and deduc- 
tive proof from a set of postulates was first made, but it was 
sharply recognized by the Greek mathematicians as early as 
550 b.c. As will appear later, there may be some grounds for 
believing that the Egyptians and the Babylonians of about 2000 
B.c. had recognized the necessity for deductive proof. For 
proof in even the rough and unready calculations of daily life 
is indeed a necessity, as may be seen from the mensuration of 

If a rectangle is 2 feet broad and 3 long, an easy proof sus- 
tains the verdict of experience, founded on direct measurement, 
that the area is 6 square feet. But if the breadth is ‘\j2 and the 
length "\/3 feet, the area cannot be determined as before by 
cutting the rectangle into unit squares; and it is a profoundly 
difficult problem to prove that the area is V6 feet, or even to 
give intelligible, usable meanings to \^2, VI j '\J6, and ‘area.’ 
By taking smaller and smaller squares as unit areas, closer and 
closer approximations to the area are obtained, but a barrier is 
soon reached beyond which direct measurement cannot proceed. 
This raises a question of cardinal importance for a just under- 
standing of the development of all mathematics, both pure and 

Continuing with the \^2 X VI rectangle, we shall suppose 
that refined measurement has given 2.4494897 as the area. This 
is correct to the seventh decimal, but it is not right, because ‘\J6, 
the exact area, is not expressible as a terminated decimal frac- 
tion. If seven-place accuracy is the utmost demanded, the area 
has been found. This degree of precision suffices for many 
practical applications, including precise surveying. But it is 
inadequate for others, such as some in the physical sciences and 
modern statistics. And before the seven-place approximation 



can be used intelligently, its order of error must be ascertained. 
Direct measurement cannot enlighten us; for after a certain 
limit, quickly passed, all measurements blur in a common uncer- 
tainty. Some universal agreement on what is meant by the exact 
area must be reached before progress is possible. Experience, 
both practical and theoretical, has shown that a consistent and 
useful mensuration of rectangles is obtained when the rule 
‘length times breadth’ is deduced from postulates abstracted 
from a lower level of experience and accepted as valid. The last 
is the methodology of all mathematics. 

Mathematicians insist on deductive proof for practically 
workable rules obtained inductively because they know that 
analogies between phenomena at different levels of experience 
are not to be accepted at their face value. Deductive reasoning 
is the only means yet devised for isolating and examining hidden 
assumptions, and for following the subtle implications of hypoth- 
eses which may be less factual than they seem. In its modern 
technical uses of the deductive method, mathematics employs 
much sharper tools than those of the traditional logic inherited 
from ancient and medieval times. 

Proof is insisted upon for another eminently practical reason. 
The difficult technology of today is likely to become the easy 
routine of tomorrow; and a vague guess about the order of 
magnitude of an unavoidable error in measurement is worthless 
in the technological precision demanded by modern civilization. 
Working technologists cannot be skilled mathematicians. But 
unless the rules these men apply in their technologies have been 
certified mathematically and scientifically by competent experts, 
they are too dangerous for use. 

There is still another important social reason for insistence 
on mathematical demonstration, as may be seen again from the 
early history of surveying. In ancient Egypt, the primitive 
theory of land measurement, without which the practice would 
have been more crudely wasteful than it actually was, sufficed 
for the economy of the time. Crude both practically and theoreti- 
cally thor ~ti this surveying was, it taxed the intelligence of the 
Egyptian, mathematicians. Today the routine of precise survey- 
ing can be mastered by a boy of seventeen; and those applica- 
tions of the trigonometry that evolved from primitive surveying 
and astronomy which are of greatest significance in our own 
civilization have no connection with surveying. Some concern 
mechanics and electrical technology, others, the most advanced 



parts of the physical sciences from which the industries of twenty 
or a hundred years hence may evolve.^, 

Now, contrary to what might be supposed, modern trigonom- 
etry did not develop in response to any practical need. Modern 
trigonometry is impossible without the calculus and the mathe- 
matics of V~l- To cite but one of the commoner applications, 
over a century and a half elapsed before this trigonometry 
became indispensable in the theory and practice of alternating 
currents. Long before anyone had dreamed of an electric dynamo, 
the necessary mathematics of dynamo design was available. It 
had developed largely because the analysts of the eighteenth 
century sought to understand mathematically the somewhat 
meager legacy of trigonometry bequeathed them by the astron- 
omers of ancient Greece, the Hindus, and the mathematicians 
of Islam. Neither astronomy nor any other science of th e eight- 
eenth century suggested the introduction of V~l> which 
completed trigonometry, as no such science ever made any use of 
the finished product. 

The importance of mathematics, from Babylon and Egypt 
to the present, as the primary source of workable approximations 
to the complexites of daily life is generally appreciated. In fact, 
a mathematician might believe it is almost too generally appre- 
ciated. It has been preached at the public, in school and out, by 
socially conscious educators until almost anyone may be par- 
doned for believing that the rule of life is rule of thumb. Because 
routine surveying, say, requires only mediocre intelligence, and 
because surveying is a minor department of applied mathe- 
matics, therefore only that mathematics which can be manipu- 
lated by rather ordinary people is of any social value. But no 
growing economy can be sustained by rule of thumb. If new 
applications of a furiously expanding science are to be possible, 
difficult and abstruse mathematical theories far beyond the 
college level must continue to be dcveloped,by those having the 
requisite talents. In this living mathematics it is imagination 
and rigorous proof which count, not the numerical accuracy of 
the machine shop or the computing laboratory. 

A familiar example from common things will show the neces- 
sity for mathematics as distinguished from calculation. A 
nautical almanac is one of the indispensablcs of modern naviga- 
tion and hence of commerce. Machines are now commonly used 
for the heavy labor of computing. Ultimately the computations 
depend upon the motions of the planets, and these are calculated 



from the infinite (non-terminating) series of numbers given by 
the Newtonian theory of gravitation. For the actual work of 
computation a machine is superior to any human brain; but no 
machine yet invented has had brains enough to reject nonsense 
fed into it. From a grotesquely absurd set of data the best of 
machines will return a final computation that looks as reasonable 
as any other. Unless the series used in dynamical astronomy 
converge to definite limiting numbers (asymptotic series also 
are used, but not properly divergent), it is futile to calculate by 
means of them. A table computed by properly divergent series 
would be indistinguishable to the untrained eye from any other; 
but the aviator trusting it for a flight from Boston to New York 
might arrive at the North Pole. Despite its inerrant accuracy 
and attractive appearance, even the most highly polished 
mechanism is no substitute for brains. The research mathe- 
matician and the scientific engineer supply the brains; the 
machine does the rest. 

Nobody with a grain of common sense would demand a strict 
proof for every tentative application of complicated mathe- 
matics to new situations. Occasionally in problems of excessive 
difficulty, like some of those in nuclear physics, calculations are 
performed blindly without reference to mathematical validity; 
but even the boldest calculator trusts that his temerity will 
some day be certified rationally. This is a task for the mathe- 
maticians, not for the scientists. And if science is to be more than 
a midden of uncorrelated facts, the task must be carried through. 

Necessity for abstractness 

With the recognition that strict deductive reasoning has both 
practical and aesthetic values, mathematics began to emerge 
some six centuries before the Christian era. The emergence was 
complete when human beings realized that common experience 
is too complex for accurate description. 

Again it is not known when or where this conclusion was first 
reached, but the Greek geometers of the fourth century b.c. 
at latest had accepted it, as is shown by their work. Thus Euclid 
in that century stated the familiar definition: “ A circle is a plane 
figure contained by one line, called the circumference, and is such 
that all straight lines drawn from a certain point, called the 
center, within the figure to the circumference are equal.” 

There is no record of any such figure as Euclid’s circle ever 
having been observed by any human being. Yet Euclid’s ideal 



circle is not only that of school geometry, but is also the circle of 
the handbooks used by engineers in calculating the performance 
of machines. Euclid’s mathematical circle is the outcome of a 
deliberate simplification and abstraction of observed disks, like 
the full moon’s, which appear ‘circular’ to unaided vision. 

This abstracting of common experience is one of the principal 
sources of the utility of mathematics and the secret of its scien- 
tific power. The world that impinges on the senses of all but 
introverted solipsists is too intricate for any exact description yet 
imagined by human beings. By abstracting and simplifying the 
evidence of the senses, mathematics brings the worlds of science 
and daily life into focus with our myopic comprehension, and 
makes possible a rational description of our experiences which 
accords remarkably "well with observation. 

Abstractness, sometimes hurled as a reproach at mathe- 
matics, is its chief glory and its surest title to practical useful- 
ness. It is also the source of such beauty as may spring from 

History and proof 

In any account of the development of mathematics there is a 
peculiar difficulty, exemplified in the two following assertions, 
about many statements concerning proof. 

( A ) It is proved in Proposition 47, Book 1, of Euclid’s Ele- 
ments, that the square on the longest side of a right-angled tri- 
angle is equal to the sum of the squares on the other two sides 
(the so-called Pythagorean theorem). 

( B ) Euclid proved the Pythagorean theorem in Proposition 
47 of Book I of his Elements. 

In ordinary discourse, ( A ), ( B ) would usually be considered 
equivalent — both true or both false. Here ( A ) is false and ( B ) 
true. For a clear understanding of the development of mathe- 
matics it is important to see that this distinction is not a quibble. 
It is also essential to recognize that comprehension here is more 
important than knowing the date (c. 330-320 b.c.) at which the 
Elements were written, or any other detail of equal antiquarian 
interest. In short, the crux of the matter is mathematics, which is 
at least as important as history, even in histories of mathematics. 

The statement ( A) is false because the attempted proof in the 
Elements is invalid. The attempt is vitiated by tacit assumptions 
that Euclid ignored in laying down the postulates from which he 
undertook to deduce the theorems in his geometry. From those 


same postulates it is easy to deduce, by irrefragable logic, spec- 
tacularly paradoxical consequences, such as “all triangles are 
equilateral.” Thus when an eminent scholar of Greek mathe- 
matics asserts that owing to the “inerring logic” of the Greeks, 
“there has been no need to reconstruct, still less to reject as 
unsound, any essential part of their doctrine,” mathematicians 
must qualify assent by referring to the evidence. The “essential 
part of their doctrine” has indeed come down to us unchanged, 
that part being insistence on deductive proof. But in the specific 
instance of Euclid’s proofs, many have been demolished in detail, 
and it would be easy to destroy more were it worth the trouble. 

The statement (B) is true because the validity of a proof is 
a function of time. The standard of mathematical proof has risen 
steadily since 1821, and finality is no longer sought or desired. 
In Euclid’s day, and for centuries thereafter, the attempted 
proof of the Pythagorean proposition satisfied all the current 
requirements of logical and mathematical rigor. A sound proof 
today does not differ greatly in outward appearance from 
Euclid’s; but if we inspect the postulates required to validate the 
proof, we notice several which Euclid overlooked. A carefully 
taught child of fourteen today can easily detect fatal omissions 
in many of the demonstrations in elementary geometry accepted 
as sound less than fifty years ago. 

It is clear that we must have some convention regarding 
‘proof.’ Otherwise, few historical statements about mathematics 
will have any meaning. Whenever in the sequel it is stated that a 
certain result was proved, this is to be understood for the sense 
as in ( B ), namely, that the proof was accepted as valid by 
professional mathematicians at the time it was given. If, for 
example, it is asserted that a work of Newton or of Euler contains 
a proof of the binomial theorem for exponents other than positive 
integers, the assertion is false for the ( A ) meaning, true for the 
(. B ). The proofs which these great mathematicians gave in the 
seventeenth and eighteenth centuries were valid at that time, 
although they would not be accepted today by a competent 
teacher from a student in the first college course. 

It need scarcely be remarked that few modest mathemati- 
cians today expect all of their own proofs to survive the criticisms 
of their successors unscathed. Mathematics thrives on intelli- 
gent criticism, and it is no disparagement of the great work of 
the past to point out that its very defects have inspired work as 



Failure to observe that mathematical validity depends upon 
its epoch may generate scholarly but vacuous disputes over 
historical minutiae. Thus a meticulous historian who asserts that 
the Greeks of Euclid’s time failed to solve quadratic equations 
by their geometric method because they ‘overlooked’ possible 
negative roots, to say nothing of imaginaries, himself overlooks 
one of the most interesting phenomena in the entire history of 

Until positive rational fractions and negative numbers were 
invented by mathematicians (or ‘discovered,’ if the inventors 
happened to be Platonic realists), a quadratic equation with 
rational integer coefficients had precisely one root, or precisely 
two, or precisely none. A Babylonian of a sufficiently remote 
century who gave 4 as the root of x- — x + 12 had solved his 
equation completely, because — 3, which we now say is the other 
root, did not exist for him. Negative numbers were not in his 
number system. The successive enlargements of the number 
system necessary to provide all algebraic equations with roots 
equal in number to the respective degrees of the equations was 
one of the outstanding landmarks in mathematical progress, and 
it took about four thousand years of civilized mathematics to 
establish it. The final necessary extension was delayed till the 
nineteenth century. 

An educated algebraist today, wishing to surpass the meticu- 
lous critic in pedantry, would point out that “how many roots 
has x 2 = x ?” is a meaningless question until the domain in 
which the roots may lie has been specified. If the domain is that 
of complex numbers, this equation has precisely two roots, 0, 1. 
But if the domain is that of Boolean algebra, this same quadratic 
(since 1847) has had n roots where « is any integer equal to, or 
greater than, 2. Boolean algebra, it may be remarked, is as legiti- 
mately a province of algebra today as is the theory of quadratic 
equations in elementary schoolbooks. In short, criticizing our 
predecessors because they completely solved their problems 
within the limitations which they themselves imposed is as 
pointless as deploring our own inability to imagine the mathe- 
matics of seven thousand years hence. 

Some of the most significant episodes in the entire history of 
mathematics will be missed unless this dependence of validity 
upon time is kept in mind as we proceed. In ancient Greece, for 
example, the entire development of by far the greater part of 
such Greek mathematics as is still of vital interest stems from 


this fact. The discontinuities in the time curve of acceptable 
proof, where standards of rigor changed abruptly, are perhaps 
the points of greatest interest in the development of mathe- 
matics. The four most abrupt appear to have been in Greece in 
the fifth century b.c., in Europe in the 1820’s and in the 1870’s, 
and again in Europe in the twentieth century. 

None of this implies that mathematics is a shifting quicksand. 
Mathematics is as stable and as firmly grounded as anything in 
human experience, and far more so than most things. Euclid’s 
Proposition I, 47 stands, as it has stood for over 2,200 years. 
Under the proper assumptions it has been rigorously proved. Our 
successors may detect flaws in our reasoning and create new 
mathematics in their efforts to construct a proof satisfying to 
themselves. But unless the whole process of mathematical devel- 
opment suffers a violent mutation, there will remain some 
proposition recognizably like that which Euclid proved in his 

Not all of the mathematics of the past has survived, even in 
suitably modernized form. Much has been discarded as trivial, 
inadequate, or cumbersome, and some has been buried as defi- 
nitely fallacious. There could be no falser picture of mathematics 
than that of “the science which has never had to retrace a step.” 
If that were true, mathematics would be the one perfect achieve- 
ment of a race admittedly incapable of perfection. Instead of this 
absurdity, we shall endeavor to portray mathematics as the 
constantly growing, human thing that it is, advancing in spite of 
its errors and partly because of them. 

Five streams 

The picture will be clearer if its main outlines are first roughly 
blocked in and retained while details are being inspected. 

Into the two main streams of number and form flowed many 
tributaries. At first mere trickles, some quickly swelled to the 
dignity of independent rivers. Two in particular influenced the 
whole course of mathematics from almost the earliest recorded 
history to the twentieth century. Counting by the natural num- 
bers 1, 2, 3, . . . introduced mathematicians to the concept of 
discreteness. The invention of irrational numbers, such as A/2, 
V3, V6; attempts to compute plane areas bounded by curves or 
by incommensurable straight lines; the like for surfaces and 
volumes; also a long struggle to give a coherent account of 


motion, growth, and other sensually continuous change, forced 
mathematicians to invent the concept of continuity. 

The whole of mathematical history may be interpreted as a 
battle for supremacy between these two concepts. This conflict 
may be but an echo of the older strife so prominent in early Greek 
philosophy, the struggle of the One to subdue the Many. But the 
image of a battle is not wholly appropriate, in mathematics at 
least, as the continuous and the discrete have frequently helped 
one another to progress. 

One type of mathematical mind prefers the problems asso- 
ciated with continuity. Geometers, analysts, and appliers of 
mathematics to science and technology are of this type. The 
complementary type, preferring discreteness, takes naturally to 
the theory of numbers in all its ramifications, to algebra, and to 
mathematical logic. No sharp line divides the two, and the 
master mathematicians have worked with equal ease in both the 
continuous and the discrete. 

In addition to number, form, discreteness, and continuity, 
a fifth stream has been of capital importance in mathematical 
history, especially since the seventeenth century. As the sciences, 
beginning with astronomy and engineering in ancient times and 
ending with biology, psychology, and sociology in our own, be- 
came more and more exact, they made constantly increasing 
demands on mathematical inventiveness, and were mainly 
responsible for a large part of the enormous expansion of all 
mathematics since 1637. Again, as industry and invention be- 
came increasingly scientific after the industrial revolution of the 
late eighteenth and early nineteenth centuries, they too stimu- 
lated mathematical creation, often posing problems beyond the 
existing resources of mathematics. A current instance is the 
problem of turbulent flow, of the first importance in aerody- 
namics. Here, as in many similar situations, attempts to solve 
an essentially new technological problem have led to further ex- 
pansions of pure mathematics. 

\yS'The timescale 

It will be well to have some idea of the distribution of mathe- 
matics in time before looking at^Ynjividual advances. 

The time curve of mathenhatical productivity is roughly 
similar to the exponential cum - of biologic growth, starting to 
rise almost imperceptibly in the Wmote past and shooting up 


with ever greater rapidity as the present is approached- The 
curve is by no means smooth; for, like art, mathematics has had 
its depressions. There was a deep one in the Middle Ages, owing 
to the mathematical barbarism of Europe being only partly 
balanced by the Moslem civilization, itself (mathematically) a 
sharp recession from the great epoch (third century b.c.) of 
Archimedes. But in spite of depressions, the general trend from 
the past to the present has been in the upward direction of a 
steady increase of valid mathematics. 

■f We should not expect the curve for mathematics to follow 
those of other civilized activities, say art and music, too closely. 
Masterpieces of sculpture once shattered are difficult to restore 
or even to remember. The greater ideas of mathematics survive 
and are carried along in the continual flow, permanent additions 
immune to the accidents of fashion. Being expressed in the one 
universally intelligible language as yet devised by human beings, 
the creations of mathematics are independent of national taste, 
as those of literature are not. Who today except a few scholars 
is interested or amused by the ancient Egyptian novelette of the 
two thieves? And how many can understand hieroglyphics suf- 
ficiently to elicit from the story whatever significance it may 
once have had for a people dead all of three thousand years ? But 
tell any engineer, or any schoolboy who has had some mensura- 
tion, the Egyptian rule for the volume of a truncated square 
pyramid, and he will recognize it instantly. Not only are the 
valid creations of mathematics preserved; their mere presence in 
the stream of progress induces new currents of mathematical 

jThe majority of working mathematicians acquainted in some 
measure with the mathematics created since 1800 agree that the 
time curve rises more sharply thereafter than before. An open 
mind on this question is necessary for anyone wishing to see 
mathematical history as the majority of mathematicians see it. 
Many who have no firsthand knowledge of living mathematics 
beyond the calculus believe on grossly inadequate evidence that 
mathematics experienced its golden age in some more or less re- 
mote past. Mathematicians, think not. The recent era, beginning 
in the nineteenth century, is usually regarded as the golden age 
by those personally conversant with mathematics and at least 
some of its history. 

An unorthodox but reasonable apportionment of the time- 
scale of mathematical development cuts all history into three 



periods of unequal lengths. These may be called the remote, the 
middle, and the recent. The remote extends from the earliest 
times of which we have reliable knowledge to a.d. 1637, the mid- 
dle from 1638 to 1800. The recent period, that of modern mathe- 
matics as professionals today understand mathematics, extends 
from 1801 to the present. Some might prefer 1821 instead of 

There are definite reasons for the precise dates. Geometry 
became analytic in 1637 with the publication of Descartes’ mas- 
terpiece. About half a century later the calculus of Newton and 
Leibniz, also the dynamics of Galileo and Newton, began to 
become the common property of all creative mathematicians. 
Le'bniz certainly was competent to estimate the magnitude of 
this advance. He is reported to have said that, of all mathematics 
from the beginning of the world to the time of Newton, •what 
Newton had done was much the better half. 

The eighteenth century exploited the methods of Descartes, 
Newton, and Leibniz in all departments of mathematics as they 
then existed. Perhaps the most significant feature of this century 
was the beginning of the abstract, completely general attack. 
Although adequate realization of the power of the abstract 
method was delayed till the twentieth century, there are notable 
anticipations in Lagrange’s work on algebraic equations and, 
above all, in his analytic mechanics. In the latter, a direct, uni- 
versal method unified mechanics as it then was, and has re- 
mained to this day one of the most powerful tools in the physical 
sciences. There was nothing like this before Lagrange. 

The last date, 1801, marks the beginning of a new era of 
unprecedented inventiveness, opening with the publication of 
Gauss’ masterpiece. The alternative, 1821, is the year in which 
Cauchy began the first satisfactory treatment of the differential 
and integral calculus. 

As one instance of the greatly accelerated productivity in the 
nineteenth century, consequent to a thorough mastery and 
amplification of the methods devised in the middle period, an 
episode in the development of geometry is typical. Each of five 
men — Lobachewsky, Bolyai, Plucker, Riemann, Lie — invented 
as part of his lifework as much (or more) new geometry as was 
created by all the Greek mathematicians in the two or three 
centuries of their greatest activity. There are good grounds for 
the frequent assertion that the nineteenth century alone contrib- 
uted about five times as much to mathematics as had all pre- 


ceding history. This applies not only to quantity but, what is of 
incomparably greater importance, to power. 

Granting that the mathematicians before the middle period 
may have encountered the difficulties attendant on all pioneer- 
ing, we need not magnify their great achievements to universe- 
filling proportions. It must be remembered that the advances of 
the recent period have swept up and included nearly all the valid 
mathematics that preceded 1800 as very special instances of 
general theories and methods. Of course nobody who works in 
mathematics believes that our age has reached the end, as La- 
grange thought his had just before the great outburst of the 
recent period. But this does not alter the fact that most of our 
predecessors did reach very definite ends, as we too no doubt 
shall. Their limited methods precluded further significant prog- 
ress, and it is possible, let us hope probable, that a century 
hence our own more powerful methods will have given place to 
others yet more powerful. 

Seven periods 

A more conventional division of the time-scale separates all 
mathematical history into seven periods : 

(1) From the earliest times to ancient Babylonia and 'Egypt, 

(2) The Greek contribution, about 600 b.c. to about a.d. 
300, the best being in the fourth and third centuries b.c. 

(3) The oriental and Semitic peoples — Hindus, Chinese, Per- 
sians, Moslems, Jews, etc., partly before, partly after (2), and 
extending to (4). 

(4) Europe during the Renaissance and the Reformation, 
roughly the fifteenth and sixteenth centuries. 

(5) The seventeenth and eighteenth centuries. 

(6) The nineteenth century. 

(7) The twentieth century. 

This division follows loosely the general development of 
Western civilization and its indebtedness to the Near East. 
Possibly (6), (7) are only one, although profoundly significant 
new trends became evident shortly after 1900. In the sequel, we 
Ghall observe what appears to have been the main contribution 
in each of the seven periods. A few anticipatory remarks here 
may clarify the picture for those seeing it for the first time. 

Although the peoples of the Near East were more active than 
the Europeans during the third of the seven periods, mathe- 



matics as it exists today is predominantly a product of Western 
civilization. Ancient advances in China, for example, either did 
not enter the general stream or did so by commerce not yet 
traced. Even such definite techniques as were devised either 
belong to the trivia of mathematics or were withheld from 
European mathematicians until long after their demonstrably 
independent invention in Europe. For example, Horner’s method 
for the numerical solution of equations may have been known to 
the Chinese, but Horner did not know that it was. And, as a 
matter of fact, mathematics would not be much the poorer if 
neither the Chinese nor Horner had ever hit on the method. 

European mathematics followed a course approximately 
parallel to that of the general culture in the several countries. 
Thus the narrowly practical civilization of ancient Rome con- 
tributed nothing to mathematics; when Italy was great in art, it 
excelled in algebra; when the last surge of the Elizabethan age 
in England had spent itself, supremacy in mathematics passed 
to Switzerland and France. Frequently, however, there were 
sporadic outbursts of isolated genius in politically minor coun- 
tries, as in the independent creation of non-Euclidean geometry 
in Hungary in the early nineteenth century. Sudden upsurges 
of national vitality were occasionally accompanied by increased 
mathematical activity, as in the Napoleonic wars following the 
French Revolution, also in Germany after the disturbances of 
1848. But the world war of 1914—18 appears to have been a 
brake on mathematical progress in Europe and to a lesser degree 
elsewhere, as also were the subsequent manifestations of na- 
tionalism in Russia, Germany, and Italy. These events hastened 
the rapid progress which mathematics had been making since 
about 1890 in the United States of America, thrusting that coun- 
try into a leading position. 

The correlation between mathematical excellence and bril- 
liance in other aspects of general culture was sometimes negative. 
Several instances might be given; the most important for the 
development of mathematics falls in the Middle Ages. When 
Gothic architecture and Christian civilization were at their 
zenith in the twelfth century (some would say in the thirteenth), 
European mathematics was just beginning the ascent from its 
nadir. It will be extremely interesting to historians eight cen- 
turies hence if it shall appear that the official disrepute into 
which mathematics and impartial science had fallen in certain 
European countries some years before the triumph of medieval 


ideals in September, 1939, was the dawn of a new faith about 
to enshrine itself in the unmathematical simplicities of a science- 
less architecture. Our shaggy ancestors got along for hundreds 
of thousands of years without science or mathematics in their 
filthy caves, and there is no obvious reason why our brutalized 
descendants — if they are to be such — should not do the same. 

Attending here only to acquisitions of the very first magni- 
tude in all seven of the periods, we may signalize three. All will 
be Rioted in some detail later. 

KJ The most enduringly influential contribution to mathematics 
of all the periods prior to the Renaissance was the Greek inven- 
tion of strict deductive reasoning. Next in mathematical impor- 
tance is the Italian and French development of symbolic algebra 
during the Renaissance. The Hindus of the seventh to the 
twelfth centuries a.d. had almost invented algebraic symbolism; 
the Moslems reverted in their classic age to an almost completely 
rhetorical algebra. The third major advance has already been 
indicated, but may be emphasized here: in the earlier part of 
the fifth period — seventeenth century — the three main streams 
of number, form, and continuity united. This generated the 
calculus and mathematical analysis in general; it also trans- 
formed geometry and made possible the later creation of the 
higher spaces necessary for modern applied mathematics. The 
leaders here were French, English, and German. 

• The fifth period is usually considered as the fountainhead of 
modern pure mathematics. It brackets the beginning of modern 
science; and another major advance was the extensive applica- 
tion of the newly created pure mathematics to dynamical 
astronomy, following the work of Newton, and, a little later, to 
the physical sciences, following the methodology of Galileo and 
Newton. Finally, in the nineteenth century, the great river 
burst its banks, deluging wildernesses where no mathematics 
had flourished and making them fruitful. . 

If the mathematics of the twentieth century differs signifi- 
cantly from that of nineteenth, possibly the most important dis- 
tinctions are a marked increase in abstractness with a consequent 
gain in generality, and a growing preoccupation with the mor- 
phology and comparative anatomy of mathematical structures; 
a sharpening of critical insight; and a dawning recognition of the 
limitations of classical deductive reasoning. If ‘limitations’ sug- 
gests frustration after about seven thousand years of human 
strivings to think clearly, the suggestion is misleading. But it is 



true that the critical evaluations of accepted mathematical 
reasoning which distinguished the first four decades of the 
twentieth century necessitated extensive revisions of earlier 
mathematics, and inspired much new work of profound interest 
for both mathematics and epistemology. They also led to what 
appeared to be the final abandonment of the theory that mathe- 
matics is an image of the Eternal Truth. 

. The division of mathematical history into about seven 
periods is more or less traditional and undoubtedly is illuminat- 
ing, especially in relation to the fluctuating light which we call 
civilization. But the unorthodox remote, middle, and recent 
periods, described earlier, seem to give a truer presentation of 
the development of mathematics itself and a more vivid sugges- 
tion of its innate vitality. • 

Some general characteristics 

In each of the seven periods there was a well-defined rise to 
maturity and a subsequent decline in each of several limited 
modes of mathematical thought. Without fertilization by crea- 
tive new ideas, each was doomed to sterility. In the Greek period, 
for example, synthetic metric geometry, as a method, got as far 
as seems humanly possible with our present mental equipment. 
It was revivified into something new by the ideas of analytic 
geometry in the seventeenth century, by those of projective 
geometry in the seventeenth and nineteenth centuries, and 
finally, in the eighteenth and nineteenth centuries, by those of 
differential geometry. 

Such revitalizations were necessary not only for the con- 
tinued growth of mathematics but also for the development of 
science. Thus it would be impossible for mathematicians to 
apprehend the subtle complexities of the geometries applied to 
modern science by the methods of Euclid and Apollonius. And in 
pure mathematics, much of the geometry of the nineteenth 
century was thrust aside by the more vigorous geometries of 
abstract spaces and the non-Riemannian geometries developed 
in the twentieth. Considerably less than forty years after the 
close of the nineteenth century, some of the geometrical master- 
pieces of that heroic age of geometry were already beginning to 
seem otiose and antiquated. This appears to be the case for much 
of classical differential geometry and synthetic projective geome- 
try. If mathematics continues to advance, the new geometries 


of the twentieth century will likely be displaced in their turn, 
or be subsumed under still rarer abstractions. In mathematics, 
of all places, finality is a chimera. Its rare appearances are 
witnessed only by the mathematically dead. 

As a period closes, there is a tendency to overelaboration of 
merely difficult things which the succeeding period either ignores 
as unlikely to be of lasting value, or includes as exercises in more 
powerful methods. Thus a host of special curves investigated 
with astonishing vigor and enthusiasm by the early masters of 
analytic geometry live, if at all, only as problems in elementary 
textbooks. Perhaps the most extensive of all mathematical 
cemeteries are the treatises which perpetuate artificially difficult 
problems in mechanics to be worked as if Lagrange, Hamilton, 
and Jacobi had never lived. 

Again, as we approach the present, new provinces of mathe- 
matics are more and more rapidly stripped of their superficial 
riches, leaving only a hypothetical mother lode to be sought 
by the better-equipped prospectors of a later generation. The 
law of diminishing returns operates here in mathematics as in 
economics: without the introduction of radically new improve- 
ments in method, the income does not balance the outgo. A 
conspicuous example is the highly developed theory of algebraic 
invariants, one of the major acquisitions of the nineteenth cen- 
tury; another, the classical theory of multiply periodic functions, 
of the same century. The first of these contributed indirectly 
to the emergence of general relativity; the second inspired much 
work in analysis and algebraic geometry. 

A last phenomenon of the entire development may be noted. 
At first the mathematical disciplines were not sharply defined. 
As knowledge increased, individual subjects split off from the 
parent mass and became autonomous. Later, some were over- 
taken and reabsorbed in vaster generalizations of the mass from 
which they had sprung. Thus trigonometry issued from survey- 
ing, astronomy, and geometry only to be absorbed, centuries 
later, in the analysis which had generalized geometry. 

This recurrent escape and recapture has inspired some to 
dream of a final, unified mathematics which shall embrace all. 
Early in the twentieth century it was believed by some for a 
time that the desired unification had been achieved in mathe- 
matical logic. But mathematics, too irrepressibly creative to be 
restrained by any formalism, escaped. 



Motivation in mathematics 

Several items in the foregoing prospectus suggest that much 
of the impulse behind mathematics has been economic. In the 
third and fourth decades of the twentieth century, for obvious 
political reasons, attempts were made to show that all vital 
mathematics, particularly in applications, is of economic 

To overemphasize the immediately practical in the develop- 
ment of mathematics at the expense of sheer intellectual curiosity 
is to miss at least half the fact. As any moderately competent 
mathematician whose education has not stopped short with 
the calculus and its commoner applications may verify for 
himself, it simply is not true that the economic motive has been 
more frequent than the purely intellectual in the creation of 
mathematics. This holds for practical mathematics as applied in 
commerce, including all insurance, science, and the technologies, 
as well as for those divisions of mathematics which at present are 
economically valueless. Instances might be multiplied indefi- 
nitely; four must suffice here, one from the theory of numbers, 
two from geometry, and one from algebra. 

About twenty centuries before the polygonal numbers were 
generalized, and considerably later applied to insurance and to 
statistics, in both instances through combinatorial analysis, the 
former by way of the mathematical theory of probability, their 
amusing peculiarities were extensively investigated by arithme- 
ticians without the least suspicion that far in the future these 
numbers were to prove useful in practical affairs. The polygonal 
numbers appealed to the Pythagoreans of the sixth century b.c. 
and to their bemused successors on account of the supposedly 
mystical virtues of such numbers. The impulse here might be 
called religious. Anyone familiar with the readily available 
history of these numbers and acquainted with Plato’s dialogues 
can trace for himself the thread of number mysticism from the 
crude numerology of the Pythagoreans to the Platonic doctrine 
of Ideas. None of this greatly resembles insurance or statistics, 
v.' Later mathematicians, including one of the greatest, regarded 
these numbers as legitimate objects of intellectual curiosity. 
Fermat, cofoundcr with Pascal in the seventeenth century of the 
mathematical theory of probability, and therefore one of the 
grandfathers of insurance, amused himself with the polygonal 


and figurate numbers for years before either he or Pascal ever 
dreamed of defining probability mathematically. 

As a second and somewhat hackneyed instance, the conic 
sections were substantially exhausted by the Greeks about seven- 
teen centuries before their applications to ballistics and astron- 
omy, and through the latter to navigation, were suspected. These 
applications might have been made without the Greek geometry, 
had Descartes’ analytics and Newton’s dynamics been available. 
But the fact is that by heavy borrowings from Greek conics the 
right way was first found. Again the initial motive was intellec- 
tual curiosity. 

The third instance is that of polydimensional space. In 
analytic geometry, a plane curve is represented by an equation 
containing two variables, a surface by an equation containing 
three. Cayley in 1843 transferred the language of geometry to 
systems of equations in more than three variables, thus inventing 
a geometry of any finite number of dimensions. This generaliza- 
tion was suggested directly by the formal algebra of common 
analytic geometry, and was elaborated for its intrinsic interest 
before uses for it were found in thermodynamics, statistical 
mechanics, and other departments of science, including statistics, 
both theoretical and industrial, as in applied physical chemistry. 
In passing, it may be noted that one method in statistical me- 
chanics makes incidental use of the arithmetical theory of 
partitions, which treats of such problems as determining in how 
many ways a given positive integer is a sum of positive integers. 
This theory was initiated by Euler in the eighteenth century, 
and for over 150 years was nothing but a plaything for experts 
in the perfectly useless theory of numbers. 

The fourth instance concerns abstract algebra as it has 
developed since 1910. Any modern algebraist may easily verify 
that much of his work has a main root in one of the most fantas- 
tically useless problems ever imagined by curious man, namely, 
in Fermat’s famous assertion of the seventeenth century that 
x n + y n — zP is impossible in integers x, y, z all different from 
zero if n is an integer greater than two. Some of this recent 
algebra quickly found use in the physical sciences, particularly 
in modern quantum mechanics. It was developed without any 
suspicion that it might be scientifically useful. Indeed, not one 
of the algebraists concerned was competent to make any signifi- 
cant application of his work to science, much less to foresee that 
such applications would some day be possible. As late as the 



autumn of 1925, only two or three physicists in the entire world 
had any inkling of the new channel much of physics was to 
follow in 1926 and the succeeding decade. 

Residues of epochs 

In following the development of mathematics, or of any 
science, it is essential to remember that although some partic- 
ular work may now be buried it is not necessarily dead. Each 
epoch has left a mass of detailed results, most of which are now 
of only antiquarian interest. For the remoter periods, these 
survive as curiosities in specialized histories of mathematics. For 
the middle and recent periods — since the early decades of the 
seventeenth century — innumerable theorems and even highly 
developed theories are entombed in the technical journals and 
transactions of learned societies, and are seldom if ever men- 
tioned even by professionals. The mere existence of many is all 
but forgotten. The lives of thousands of workers have gone into 
this moribund literature. In what sense do these half-forgotten 
things live r And how can it be truthfully said that the labor of 
all those toilers was not wasted ? 

The answers to these somewhat discouraging questions are 
obvious to anyone who works in mathematics. Out of all the 
uncoordinated details at last emerges a general method or a new 
concept. The method or the concept is what survives. By means 
of the general method the laborious details from which it evolved 
arc obtained uniformly and with comparative ease. The new 
concept is seen to be more significant for the whole of mathe- 
matics than are the obscure phenomena from which it was 
abstracted. But such is the nature of the human mind that it 
almost invariably takes the longest way round, shunning the 
straight road to its goal. There is no principle of least action in 
scientific discovery. Indeed, the goal in mathematics frequently 
is unperccivcd until some explorer more fortunate than his 
rivals blunders onto it in spite of his human inclination to follow 
the crookcdest path. Simplicity and directness are usually the 
last things to be attained. 

In illustration of these facts we may cite once more the 
theory of algebraic invariants. When this theory was first de- 
veloped in the nineteenth century, scores of devoted workers 
slaved at the detailed calculation of particular invariants and 
covariants. Their work is buried. But its very complexity drove 
their successors in algebra to simplicity: masses of apparently 


isolated phenomena were recognized as instances of simple 
underlying general principles. Whether these principles would 
ever have been sought, much less discovered, without the urge 
imparted by the massed calculations, is at least debatable. The 
historical fact is that they were so sought and discovered. 

In saying that the formidable lists of covariants and invar- 
iants of the early period are buried, we do not mean to imply 
that they are permanently useless; for the future of mathematics 
is as unpredictable as is that of any other social activity. But 
the methods and principles of the later period make it possible 
to obtain all such results with much greater ease should they 
ever be required, and it is a waste of time and effort today to 
add to them. 

One residue of all this vast effort is the concept of invariance. 
So far as can be seen at present, invariance is likely to be il- 
luminating in both pure and applied mathematics for many 
decades to come. In our survey we shall endeavor to observe the 
methods and the concepts which have been sublimated from 
other masses of details, and which offer similar prospects of 
endurance. It is not epochs that matter, but their residues. Nor, 
as epochs recede into the past, do the men who made them 
obscure the permanence and impersonality of their work with 
their hopes, their fears, their jealousies, and their petty quarrels. 
Some of the greatest things that were ever done in mathematics 
are wholly anonymous. We shall never know who first imagined 
the numbers 1, 2, 3, . . . , or who first perceived that a single 
‘three’ isolates what is common to three goads, three oxen, 
three gods, three altars, and three men. 

Two recent opinions on the general history of science are 
apposite for that of mathematics, and may stand here as an 
introduction to what is to follow. In his Autobiograpliia (1923), 
the Spanish histologist Santiago Ramon y Cajal had this to say 
of scientific history: 

In spite of all the allegations of self-love, the facts at first associated with 
the name of a particular man end by being anonymous, lost forever in the 
ocean of Universal Science, Thus thij monograph imbued with individual 
human quality becomes incorporated, stripped of sentimentalisms, in the ab- 
stract doctrine of the general treatise. To the hot sun of actuality will succeed 
— if they do succeed — the cold beams /si the history of learning. 

The next is singularly pdrtinent, coming as it does from the 
man who advanced beyond Newton in the mathematical theory 


of gravitation. Speaking of Newton’s w r ork in optics, Einstein 
says : 

Newton’s age has long since passed through the sieve of oblivion, the 
doubtful striving and suffering of his generation have vanished from our ken; 
the works of some few great thinkers and artists have remained, to delight 
and ennoble those who come after us. Newton’s discoveries have passed into 
the stock of accepted knowledge. 

Finally, we shall try to observe the caution suggested in the 
observation of an M.D. and writer who is not a mathematician, 
Halladay Sutherland: “There is always the danger of seeing the 
past in the light of a golden sunset.” 


The Age of Empiricism 

It is not known where, when, or by whom, it was first per- 
ceived that a mastery of number and form is as useful as lan- 
guage for civilized living. The historical record begins, in Egypt 
and in Mesopotamia (Babylonia, including Sumer and Akkad), 
with both number and form far advanced beyond the primitive 
stage of culture, and even here the cardinal dates have been 
disputed. Those dates are 4241 ± 200 b.c. at the earliest and 
2781 b.c. at the latest for Egypt, 1 and about 5700 b.c. for 
Mesopotamia. Both refer to the earliest calendric reckoning, 
and each is more or less substantiated by astronomical evidence. 

The basis of both the Egyptian and the Mesopotamian civi- 
lizations was agriculture. In an agricultural economy a reliable 
calendar is a necessity. A calendar implies both astronomical and 
arithmetical accuracy far beyond the facilities of mythology and 
haphazard observation, and it is not come at in a year. Some 
primitive peoples who have never been driven to farming have 
only the vaguest notions of the connection between the periodi- 
city of seasons and the aspect of the heavens. By 5700 b.c. the 
Sumerian predecessors of the Semitic Babylonians were dating 
the beginning of their year from the vernal equinox. A thousand 
years later the first month of the year was named after the Bull, 
the sun being in the constellation Taurus at the vernal equinox 
of about 4700 b.c. Thus the inhabitants of Mesopotamia must 
have had a workable elementary arithmetic. 

These same pioneers toward mathematics also invented or 
helped to transmit two major curses which continue to blight the 
unscientific mind, numerology (number mysticism) and astrol- 
ogy. It is an open question which of astrology or astronomy 



preceded the other. Arithmetic of some sort necessarily came 
before numerology. 

For Egypt, the early historical record is somewhat more de- 
tailed. The more liberal of rival Egyptian chronologies assigns 
4241 n.c. as the earliest precise date in history, this coinciding 
with the adoption of the Egyptian calendar of twelve thirty-day 
months with five days of feasting to complete the 365. This date 
also is supported by inconclusive astronomical evidence, correlat- 
ing the heliacal rising of the Dog Star Sothis, our Sirius, with the 
date at which the annual inundation by the Nile could be 
expected. Here again the impulse to develop astronomy, and 
hence also arithmetic, was agricultural necessity unless, of 
course, it was astrology. 

The geographical location of Sumer was more propitious than 
that of Egypt for a rapid development of the mathematics con- 
ceived in agriculture and born in astronomy. Egypt lay far 
off the main trade route between East and West. Sumer, 
the non-Semitic predecessor of Semitic Babylonia, lay directly 
across the path of the merchants at the north end of the Persian 
Gulf. Commerce stimulated mathematical invention in Sumer 
and ancient Mesopotamia as it probably never has since. Europe 
of the late Middle Ages also profited mathematically through 
trade; but the gain was in a diffusion of knowledge rather than 
in the creation of new mathematics necessitated by commerce. 

Possibly of greater importance than trade for the develop- 
ment of mathematics were the demands of primitive engineering. 
Both the Babylonians and the Egyptians were indefatigable 
builders and skilled irrigation engineers, and their extensive 
labors in these fields may have stimulated empirical calculation. 
But it would be gratuitous generosity to infer that because the 
Egyptians, say, succeeded in raising huge obelisks, they were 
therefore engineers in any sense that would now be recognized 
as scientific. Ten thousand slaves can muddle through the work 
of one head; and the apparent marvels of ancient engineering 
that impress us today may be only monuments to a lavish 
expenditure of brawn and a strict conservation of brains. The 
Israelites and others whom the Egyptians persuaded to take up 
practical engineering do not seem to have been greatly impressed 
by the technical skill of their overseers. 

Reliable evidence shows that arithmetic and mensuration in 
Babylonia developed from the early work of the non-Semitic 
Sumerians. This gifted people also invented a pictorial script 


which evolved into the efficient cuneiform characters that were to 
prove adequate for the expression of their arithmetic and mensu- 
ration. The political absorption of the Sumerians by physically 
but not intellectually more vigorous peoples occurred about 2000 
b.c. Astronomy and arithmetic continued to flourish and, what 
is of singular significance, a sort of algebra evolved with incredible 
speed. This early appearance of algebra is one of the most re- 
markable phenomena in the history of mathematics. 

For all that is known to the contrary, other early civiliza- 
tions may have made progress toward mathematics comparable 
to that of Mesopotamia and Egypt, records of these two having 
survived largely by physical accident. The semiarid climate of 
Egypt and the inordinate reverence of the Egyptians for all 
their dead, including bulls, crocodiles, cats, and human beings, 
united to preserve the papyri that must have perished in a 
harsher atmosphere, and kept the memories of common things 
colorful for thousands of years on the walls of tombs and temples. 
Some of the most interesting historical documents as yet re- 
covered from the past survived only because the Egyptian 
morticians discovered that useless papyri made excellent stuffing 
to plump out the mummies of sacred crocodiles to lifelike 

The Babylonians impressed their records on a yet more 
durable medium, clay tablets, cylinders, and prisms, baked in 
the sun or in kilns. Sharpened sticks, one like an implement still 
used by children in modeling, indented the wedge-shaped charac- 
ters in the soft clay, and the baking fixed a record more durably 
than any printers’ ink on the toughest paper. Wars and the long 
decadence of a great civilization for once conspired to preserve 
some of the best in that civilization. The baked tablets, resistant 
to damp, rust, and pressure, and immune to the attacks of worms 
and insects, were buried beneath the mud ruins of dissolving 
temples and libraries. It would be easier for some science-hating 
zealot to obliterate modern mathematics than it would be for us 
to destroy the mathematical records of Babylonia. There is no 
reason to suppose that all the mathematical bricks have been 

If the records themselves are solid and tangible beyond dis- 
pute, the like cannot be claimed for their interpretation. The 
reading of the most suggestive parts of the Sumerian and Baby- 
lonian records is a matter of great difficulty, demanding an un- 
usual combination of linguistic, historical, and mathematical 


talents. Several points of interest are still in dispute among the 
scholars who since 1929 have finally broken the seals on ancient 
Babylonian mathematics. We shall not find it necessary to use 
any of this disputed material in order to give an idea, sufficient 
for our purposes, of what the Babylonians accomplished. What 
remains after the few doubtful items are discarded is impressive 

Those far-off centuries in Babylon and Egypt are the first 
and last great age of the empiricism that led to mathematics. 
Above a multitude of details, five epochal landmarks survived 
for the guidance of later centuries. Number was subdued to the 
service of astronomy and commerce; the perception of form was 
clarified in an empirical mensuration and applied to astronomy, 
surveying, and engineering; the vast extensions of the natural 
number system which mathematics uses today were initiated; a 
method more powerful than arithmetic was begun in an algebra 
more than well begun; and last, also perhaps most significantly, 
practical difficulties in mensuration compelled some of those 
early empiricists to grapple at least subconsciously with the con- 
cept of the mathematical infinite. From that day to this, a 
stretch of nearly four thousand years, the struggle to compass 
the infinite has continued, and the record of the struggle is 
mathematical analysis. 

Possibly of greater significance for the future of the race than 
all the technical advance toward mathematics was another for 
which that advance was to be largely responsible. It dawned on 
the human mind that man might dispense •with the thousands 
of capricious deities created by human beings in the childhood 
of their race, and give a rational account of the physical universe. 
Although an explicit statement of this possibility was to be re- 
served for one of the earliest and greatest of the Greek mathe- 
maticians, it was anticipated by the astronomers and scientists 
of Egypt and Babylon, and it was there that our race began to 
grow up. 

Arithmetic- to 600 b.c. 

Since 1929 our knowledge of mathematics in ancient Baby- 
lonia has been increased many times over all that was previously 
known, largely through the pioneering work of O. Neugebauer 
(1S99-). Apart from their great intrinsic interest, these new 
accessions are extremely suggestive as possible clues to the 


the sexagesimal (60 as base), with a slight admixture of the 
decimal system with 10 as base. 

It has been conjectured that the 10 commemorates finger 
counting, while the 60 is 6 X 10, the 6 being adopted so that 
the useful fractions l/2 a 3 b 5 e ( a , b, c non-negative integers) would 
be expressible in finite terms. Traces of the sexagesimal system 
survive in our reckoning of time and in the corresponding divi- 
sion of the circumference of a circle into 6 X 60 degrees. But it 
is no longer universally supposed that such considerations in- 
duced the Sumerians to choose 60 as a base, and still less that 
the zodiac influenced their choice. 

The place-value system was used for both positive and nega- 
tive powers of the base. Thus, in the appropriate cuneiform 
symbols, 17, 35; 6, 1, 43, the semicolon indicating the beginning 
of the fractional part, denotes 17 X 60 + 35 -f- 6/60 -f 1/60 2 
-f-43/60 3 . The method of writing sometimes introduces am- 
biguities; but the pertinacity of modern scholarship has removed 
these and made sense of the residue. According to the mood of 
the scribe, a blank might or might not indicate absence of the 
corresponding power of 60. This particular difficulty was over- 
come by the introduction of a special character for zero but not, 
probably, until the time of the Greeks. 

The great practical invention of zero has usually been attrib- 
uted to the Hindus, and it may still be debatable whether they 
or the Babylonians were first. If, as seems highly probable, the 
Babylonians were original in their invention, zero is an interest- 
ing example of the independent origins of mathematical ideas in 
different cultures. Zero also appeared in the arithmetic of another 
gifted people, the Mayas of Central America, who used 20 as a 
base and had a system of place-value. The Maya numeration 
has been assigned to the period a.d. 200-600. Their calcndric 
cycles go back to 3373 b.c., but this does not imply that the 
Mayas were civilized or even in existence that early. 

The Babylonians were among the most indefatigable com- 
pilers of arithmetical tables in history. Since it was easier to 
multiply than to divide, they tabulated 1/n for integers n 
adapted to the base 60. Other reciprocals, ‘irregulars,’ like i, -Jy, 
naturally caused more trouble, but were competently avoided 
by manufacturing problems in which such awkward divisors 
would automatically drop out in the course of the work. This is 
not the only instance in Babylonian mathematics where the 
teacher or the pupil appears to have applied a technique once 


classical in mathematical physics: given the solution, find the 
problem. There were also multiplication tables for such multi- 
pliers as 7, 10, 12^, 16, 24, etc. Tables of squares appropriately 
read served as tables of square roots, and likewise for cubes. 
Another table listed values of n 3 + n 2 for n = 1, 2, . . . , 30. 
The peculiar significance of this strange tabulation -will appear 
when we come to Babylonian algebra. 

From all this and a great deal more of a similar character, it 
is evident that the Babylonians of about 2000 b.c. were highly 
skilled calculators. It may not be too generous to credit them 
with an instinct for functionality; for a function has been suc- 
cinctly defined as a table or a correspondence. 

Historically, the most remarkable thing about this rapid 
progress in the subjugation of number is that it appears to have 
been ignored by the Greeks of the sixth century b.c. For what 
now seems to us the simplest, most natural development of 
mathematics this was a calamity. The fact that it happened 
casts a slight shadow of doubt on the vaunted intelligence of the 
early Greek mind. But since to press this point would be tanta- 
mount to historical blasphemy, we merely suggest that the 
mathematically informed observer examine the evidence and 
reach his own conclusions, even at the risk of upsetting sacro- 
sanct tradition. 

Egyptian arithmetic shows even more starkly its laboriously 
empirical origin. As early as 3500 b.c., the Egyptians freely 
handled numbers in the hundreds of thousands. Their hiero- 
glyphics of this early date actually record the capture of 120,000 
human prisoners, 400,000 oxen, and 1,422,000 goats. The last is 
probably a poetic flight of the conqueror’s imagination — the 
catalogue occurs on a royal mace — for even today the experts of 
the U.S. Census Bureau would be exercised to enumerate that 
many goats in the brief interval between victory and celebra- 
tion. But the enthusiastic exaggeration, like that of the ancient 
Hindus in multiplying their gcds practically to infinity, shows at 
least that the Egyptians of 3500 b.c. had completely overcome 
the inability of primitive peoples to think boldly in terms of 
numbers. The significance of this advance can be appreciated 
only by comparison with the arithmetical backwardness of 
peoples well beyond barbarism today and also, as will appear, 
once more with the Greeks. '• 

Egyptian numeration followed the decimal system, but with- 
out place-values. The arithmetic of about 1650 b.c. was capable 



of addition, subtraction, multiplication, and division, and was 
applied to numerous extremely simple problems involving all 
these operations. In fractions, 3 was denoted by a special symbol; 
other fractions were reduced to sums of fractions of the form 
l/«, n an integer. In the Rhind papyrus of about 1650 b.c., 
copied by the scribe Ahmes (A’h-mose) from an older work, divi- 
sions are performed by means of these ‘unit fractions,’ the 
technique being the expression of m/n, in > 1 , as a sum of unit 
fractions; for example, + Trhr + rW- How such curi- 

ous resolutions were first obtained seems not to be known. They 
may represent the fossil experience of centuries carefully pre- 
served in tables for future use, as we today store up logarithms. 
Ahmes transmitted resolutions of all fractions 2 /«, where n 
is any odd number from 5 to 101. These could have been derived 
from successive applications of the solution in positive integers 
of the so-called optical equation, l/x + 1 /y = l/n, but it is 
most improbable that they were. Of the many other conjectures, 
none is acceptable to a majority of competent scholars. 

All of the problems solved arc childishly simple. Some arc 
quite delightful for their unintentional revelations of Egyptian 
manners and customs, as when Ahmes goes into the arithmetic 
of bartering beer for bread and vice versa. Either the Egyptians 
were less puritanical in their schools than we, or Ahmes intended 
his treatise only for expert mathematicians. Less inflammatory 
problems arc concerned with the rationing of oxen and various 
kinds of birds from geese to cranes and quail. A more fanciful 
kind, obviously of no earthly use to anyone, recalls the older 
type of English examination question and the more antique 
efforts of our College Entrance Examination Board. Loaves of 
bread arc partitioned among several imaginary beings who are to 
receive amounts in arithmetical progression. There is nothing 
new, provided it be silly enough, under the sun. 

The most significant detail for the development of mathe- 
matical thought in all the Egyptian arithmetic is the occasional 
checking of a calculation. This seems to show that the Egyptians 
of at latest the seventeenth century b.c. understood the value of 
proof in arithmetic. If this is a justifiable conclusion, those 
ancients were well on their way to mathematics 'when, un- 
accountably, they stopped. 

It is said that the Egyptians’ arithmetic was sufficiently 
advanced for the simple demands of their daily affairs. Of greatet 
interest, perhaps, for the evolution of mathematics are those t'ery 


century B.c. of the correct rules of signs in multiplication. How 
the Greeks overlooked all this is a mystery. 

Algebra without symbolism - 

Passing on to Babylonian algebra of about 2000 b.c., we come 
to what historians consider the most remarkable anticipation 
in the development of mathematics. 

First is the question of proof, without which mathematics in 
the accepted sense does not exist. Did the Babylonian successors 
of the Sumerians have any conception of deductive reasoning? 
No categorical answer can be given. So far (1945) there has not 
been discovered any Babylonian record of a mathematical dem- 
onstration. But this is not necessarily conclusive against at least 
a mute intuition for proof, evidence for which is overwhelming. 

To put the case as favorably as seems justified by the undis- 
puted evidence, we may picture a teacher of elementary algebra 
today grading an examination on quadratic equations. The 
pupils have been asked to solve 12* 2 — 7* = — 1. Some h ave 
substituted in the standard formula x = (— b ± \b 2 — 4ac)/2a 
for the solution of ax- -f- bx + c = 0 and have been satisfied 
with one root; others have ‘completed the square’; and one 
original genius has ‘normalized’ the equation by multiplying 
throughout by the coefficient of x 2 , getting 

(12 *) 2 - 7(12*) = -12, 

before solving for 12*, whence he easily finds * by division. 
Falling short of the better Egyptians 3,500 years before him, 
not one of the pupils has sought to verify his solution by sub- 
stituting in the equation, as the harried teacher had forgotten 
to demand a check. Nor has anyone offered a word of proof 
in support of his formal calculations. All have gone through 
the steps to a solution as if the teacher, with an open book in her 
hand, were dictating from first to last what to do next. 

Equivalents of all this, including the absence of verification, 
occur in the Babylonian tablets of about 2000 b.c. Verbal instruc- 
tions direct the solver to follow a path leading to the solution 
by our standard formula, or to normalize the equation, or to 
complete the square. It is algebra by rule and without algebraic 
symbolism. The scribes who indented these paradigms on the 
soft clay, or who directed others, certainly had general pro- 
cedures in mind. But it must be admitted that correct general 


rules, even when successfully applied to hundreds of special 
examples, do not constitute mathematical proof. Ail of the 
Babylonian algebra is of this character: detailed solution of one 
numerical problem after another by verbal instructions follow- 
ing definite patterns. No pattern is ever isolated as a general 

The empirical character of the algebra of the Babylonians, 
and perhaps also their social outlook, is even more strikingly evi- 
dent in their astonishing solutions of cubic equations with 
numerical coefficients. Expressed in our terminology, equations 
of the type x z -f- px 2 -f- q = 0 are reduced to the normal form 
y 3 -j- y 2 — vrith y as x/p, t = —q/p z , by multiplying the 
original equation throughout by 1/p 3 . If the resulting r is posi- 
tive, the value of y, and hence that of x, is obtainable from 
tabulated values of n z -f- n 2 , provided r is in the table. 

From the equations actually solved in this manner, it is con- 
ceivable that the scribe proceeded from certain tabulated ris to 
construct his equations x z -f- px 2 = q so that they would be 
solvable. Then he triumphantly produced the solution. If so, his 
pupils must have been as thoroughly mystified as is any student 
vrith a grain of mathematical intelligence today when his mys- 
terious instructor pulls mathematical rabbits out of invisible 
hats. Brilliant trickery is no longer considered reputable mathe- 
matics. But if it is true, as has been asserted, that the mathe- 
matics of Babylonia and Egypt was the jealously guarded secret 
of a priestly sect, the mystery vanishes. One of the greatest 
services the Greek mathematicians rendered civilization was 
their shattering of the tradition of secrecy fostered by self-per- 
petuating priesthoods. The attempt of Pythagoras to carry on 
the secretive tradition of Babylon and Egypt was quickly dis- 
sipated, and enlightenment was put within the grasp of any 
unsanctified vulgarian with the will and the intelligence to reach 
for it. 

It has been conjectured that the reduction of the general 
cubic to the above normal form was within the powers of the 
Babylonian algebraists. But we need not assume this much to 
grant that the Babylonians had taken a long stride toward 
mathematics in what they actually did. For the spirit animating 
mathematical discovery is the recognition of uniformity in a host 
of apparently diverse phenomena. The project of reducing a 
multitude of particular equations to a standard form, or even 
the easier inverse problem of constructing special equations 


indefinitely to fit a prescribed solution, would occur only to an 
intelligence that was essentially mathematical. 

This methodology of transformation and reduction, general- 
ized and many times refined with advancing knowledge, runs like 
a scarlet thread through all the greater epochs of mathematics. 
A relatively difficult problem is reduced by reversible transfor- 
mations to a more easily approachable one; the solution of the 
latter then drags along with it the solution of the former and of 
all problems of which it is the type. The Babylonian reduction 
of cubics appears to be the first recorded instance of this method- 
ology. We shall observe it again in the Italian algebra of the 
early sixteenth century, and in Vieta's signal advance half a 
century later. In geometry, to cite but one instance, the method 
first appears in the device of central projection, whereby geom- 
eters in the seventeenth century derived the properties of conics 
from those of the circle. 

Taking the view, as we shall in general, that uniform methods 
are of more lasting significance than the sum total of the indi- 
vidual results, however brilliant or useful, obtained up to any 
given epoch by their use, we might rest the case for the Babylo- 
nians as mathematicians on their reduction of cubics to a normal 
form. But the spectacular ingenuity of their algebra — when we 
consider that nothing surpassing it was known in Europe till the 
sixteenth century a.d. — demands a summary indication of 
certain particulars. 

Always relying on their extensive numerical tables, the Baby- 
lonian algebraists solved simultaneous linear equations in two 
unknowns, also simultaneous quadratics of the type xy *= 600, 
(fl.v + iy) 2 + cx -f- dy — e for 55 sets of special numerical values 
of a, b, c, d, c , each of the sets leading to a quadratic in x. They 
also proposed a problem leading to the general quartic which, it 
need hardly be said, they did not solve; and likewise for a general 
cubic arising from a problem on frustums of pyramids. A cubic in 
x 1 also appears. In their solutions of quadratics the Babylonians 
were usually content with one root, although in one example 
both roots (positive) are given. A multiplicity of unknowns does 
not seem to have dismayed them, one problem leading to ten 
linear equations in ten unknowns. 

Even more remarkable, perhaps, is the successful solution by 
initial trial and subsequent interpolation of an exponential 
equation to determine the time required for a sum of money to 
double itself at a stated rate of compound interest. Such equa- 


tions are solved today by logarithms. But to infer that the 
Babylonians understood logarithms even to base 2 would be 
as fantastic as that classic fable of archaeology which declares 
that the ancient Egyptians were familiar with wireless te- 
legraphy because not a scrap of wire has been found in their 

In another direction, the Babylonians partly anticipated the 
summation of a geometric progression by Archimedes in the 
third century b.c., giving the correct result for ten terms by a 
special case of the general rule. 

Of greater significance for the future of mathematics was the 
highly intelligent reaction of the Babylonian algebraists to 
irrationals. Their tables and their equations taught them that 
not every rational number in their tables had a tabulated square 
root. Faced with this fundamental fact, they proceeded to 
approximate by means of the rules ( a 2 + b 2 )* = a -f- & 2 / 2 i 2 , or 
= a 2 -\- lab 2 . The first is reasonable and reappears about two 
thousand years later with Heron of Alexandria; the second is 
hopelessly wrong, being dimensionally impossible. We note in 
passing that the reasonable approximation is obtainable from 
Newton’s binomial series; but again this does not imply anticipa- 
tion. In further approximations to quadratic surds they used 
what may be interpreted as the first steps toward conversion 
into periodic continued fractions. For they gave the approxi- 
mation ly^-, correct to two decimals. As will be seen when we 
consider the Pythagoreans, yj2 marks one of the cardinal turn- 
ing points in the history of mathematics. 

Contemplating work of this caliber, done for the most part 
about two thousand years before the Christian era, we can only 
marvel how it was done, for we do not know. The detailed 
numerical solution of specific examples gives no hint of the 
thought inspiring the uniform procedures. Neugebauer empha- 
sizes that the technique is based on elaborate numerical tables. 
At the lowest estimate, the high skill in using such tables indi- 
cates an extraordinary capacity for detecting uniformities in 
masses of empirical data. The Babylonians were the world’s 
first exact astronomers; and so accurate were their first observa- 
tions and their calculations that Kidinnu, about 340 b.c., made 
the capital discovery of the precession of the equinoxes, antici- 
pating Hipparchus by about two centuries. It seems reasonable 
to assume that unrecorded centuries of observing the planets 
led to the accumulation of numerical data from which a purely 



rhetorical algebra evolved. For Babylonian algebra -was entirely 
unsymbolic. Alore remarkable still, the processes which now 
would be summarized in formulas were never, so far as is known, 
reduced to written rules. If these elaborate procedures were 
transmitted wholly by word of mouth, the strain on even a 
strong memory must have been considerable. 

Another unsolved problem is even more puzzling. Up to 1900 
it was customary to ascribe the beginnings of algebra to the 
Greek Diophantus in the third century a.d., over two thousand 
years after the Babylonians had bettered some of his best. 
Where was algebra buried in the meantime? It has been con- 
jectured that the Greeks of the sixth and fifth centuries b.c. must 
have been acquainted with what the Babylonians had done in 
algebra because, as will appear, a considerable amount of Baby- 
lonian empirical geometry almost certainly was known to those 
same Greeks, Direct evidence is lacking that the early Greeks 
were not acquainted with Babylonian algebra, but the indirect 
evidence is at least worth noting. For if the early Greeks were 
cognizant of Babylonian algebra, they made no attempt to 
develop or even to use it, and thereby they stand convicted of 
the supreme stupidity in the history of mathematics. But it is 
commonly agreed that the early Greek mathematicians and 
philosophers were among the most intelligent human beings 
that ever lived. 

This awkward historical dilemma can be circumvented by a 
slight anticipation. The ancient Babylonians had a rare capacity 
for numerical calculation; the majority of the Greeks were either 
mystical or obtuse in their first approach to number. What the 
Greeks lacked in number, the Babylonians lacked in logic and 
geometry, and where the Babylonians fell short, the Greeks 
excelled. Only in the modern mathematical mind of the seven- 
teenth and succeeding centuries were number and form first 
clearly perceived as different aspects of one mathematics. 

Nothing has been said about Egyptian algebra because it was 
far less advanced than the (probably) earlier work of the Baby- 
lonians. Between 1850 and 1650 b.c. the Egyptians solved easy 
numerical equations of the first degree by trial, or by what was 
called the rule of false position in the Middle Ages. The last 
makes it plausible that the Egyptians understood proportion. 
If they did, and historical experts do not doubt it, they share 
with the Babylonians the honor of having uncovered a main 
root of mathematical analysis. 


Toward geometry and analysis 3 

Babylonian mensuration 2 of about 2200-2200 b.c. is almost 
as astonishing as the contemporary algebra. Mathematically, it 
is of the same character as the algebra in its disregard of proof. 
Correct rules are applied for finding the area of any rectangle, 
right triangle, isosceles triangle, trapezoid with one side per- 
pendicular to the base, and “if t be taken equal to 3,” any 
circle. This approximation to 7 r is famous also for its occurrence 
in the Old Testament. A little later, between 1850 and 1650 
b.c., the Egyptians had the closer approximation ~ 3.16. 
It would be interesting to know what suggested the curious (g-) 4 . 

In their mensuration of solids the Babylonians of about 2000 
b.c. gave correct solutions of numerical problems involving 
rectangular parallelepipeds, right circular cylinders, and right 
prisms with trapezoidal bases. Some of this had obvious applica- 
tions to earthwork problems in the excavation of canals for 
drainage or irrigation. Their rule for the volume of a truncated 
square pyramid was incorrect. The correct rule of the Egyptians, 
one of the most remarkable achievements of pre-Greek mathe- 
matics, will be considered later by itself. 

Passing to theorems of pure geometry known to the Baby- 
lonians of the same period, we select three for their outstanding 
historical suggestiveness. The first two are: the angle in a semi- 
circle is a right angle; the Pythagorean theorem c 2 = a 2 + b 2 t 
where c,a,b are the sides of a right triangle, for certain numerical 
values of c,a,b, as 20, 16, 12 and 17, 15, 8. The first of these, often 
considered one of the most beautiful theorems in elementary 
geometry, is said to have been proved by the Greek Thales about 
600 b.c. It would be guessed immediately on inscribing rectangles 
in circles. The Babylonians offered no justification. From the 
second, and certain numerical calculations for the sides of right 
triangles, it has been argued that the Babylonians knew the 
Pythagorean theorem in the general case, but the evidence 
seems inconclusive. Until 1923 it was supposed that the Egyp- 
tians knew this theorem at least in the case 5 2 = 4 2 + 3 2 , be- 
cause the Egyptian ‘rope stretchers’ were.formerly said to have 
used this property of 5, 4, 3 in laying out right angles for the 
orientation of buildings. But it is now claimed that even though 
5, 4, 3 may have been used thus, the Egyptians knew not a single 
instance of the Pythagorean c 2 = a 2 -j- b 2 , because there is no 
documentary evidence that they did. Since c, a, b are the sides 



of a right triangle if and only if c 2 = a- + b 2 , we have here an 
interesting historical puzzle as to how the Egyptians guessed 
what they needed. 

Regarding the Pythagorean theorem itself, whoever first 
guessed it, we recall that it is the cornerstone of Euclidean 
metric geometry and one of the bases of all metrics. It too, like 
similar triangles, threads all mathematical history, not only in 
geometry, but also in algebra, the theory of numbers, and mathe- 
matical physics. 

The third significant empirical theorem of the Babylonians 
in pure geometry is the earliest recorded trace of the origins of 
mathematical analysis: the sides about corresponding angles of 
similar triangles are proportional. This theorem implies equality 
of ratios. It has been said to follow that the Babylonians had 
some conception of ratio. But, if we wish to be as precise here as 
we were a moment ago in the case of the Pythagorean theorem 
versus the nonsuited Egyptians, we have no right to assert that 
the Babylonians actually had even the remotest conception of 
ratio. For 'equal ratios’ and ‘ratio’ are distinct concepts in 
mathematics, and an extensive theory of ‘equal ratios’ is easily 
possible without any definition of ‘ratio.’ ‘Ratio’ is on a higher 
level of abstraction than ‘equal ratios.’ Euclid attempted, not 
too successfully, to define ratio. His definition has been trans- 
lated by De Morgan thus: “ratio is a certain mutual habitude of 
two magnitudes of the same kind depending upon their quantu- 
plicity.” Fortunately Euclid never had to appeal to this abstruse 
definition, his ‘theory of ratio’ being wholly a theory of propor- 
tion, that is, of equal ratios. That the Babylonians, or anyone 
else before the nineteenth century, had a workable conception 
of ratio seems extremely improbable. The ratio of in to n, written 
usually as m/n, is understandable only — so far as we know at 
present — as a number-couple (?«, n) with certain postulated 
properties for the four rational operations on such couples. 
So far as documentary evidence goes, there is none, apparently, 
to show that the Babylonians ever got within hailing distance of 
Euclid, who, if he did not succeed in giving an unmystical defini- 
tion of ratio, at least acted as if he were aware that ‘ratio’ and 
‘proportion’ are different concepts. But in this whole matter we 
have no desire to be as precise as we were in the case of the 
Egyptians; we have tried merely to indicate where the mathe- 
matical crux of the history lies. With their numerical examples 
of four numbers in proportion, the Babylonians took the first 


step toward the Greek theory of proportion which has lasted, 
practically unmodified, to this day. 

Another possible source of much modern mathematics will be 
noted presently in connection with the pyramid. But the sub- 
sequent history of what evolved from similar triangles is so clear, 
and of such outstanding importance for all mathematics, that 
we shall leave Babylonian metrics here with this as its crown. 

With one exception to be discussed presently, the Egyptian 
empirical mensuration is less impressive than the Babylonian. 
From their prodigious architecture it might be reasonable to 
infer that the ancient Egyptians were skilled construction engi- 
neers and hence at least respectable geometers. They were 
neither. Brute force in the form of unlimited slave labor made 
brains all but superfluous. Until it was discovered how they 
raised the huge stone blocks to build their pyramids, it was 
supposed that the Egyptian overseers were acquainted with at 
least the rudiments of scientific engineering. What they actually 
did 4 puts them on the intellectual level of the ants. As the suc- 
cessive tiers of a pyramid rose, the slaves laboriously buried 
under thousands of tons of sand the face of the work already 
done. The swarms of slaves lugged the blocks up the long ramp. 
When the task of building was finished, the slaves removed the 
mountain of sand burying the pyramid, to put it all back where 
they had got it in the first place. The dazzling result of their 
labors shone out in all its splendor, another time-outlasting 
monument to the unconquerable spirit of man’s temporal rulers 
and the unbreakable backs of those who do the work. The sagac- 
ity of the slave-driving Egyptian overseers has been rated highly 
by competent enthusiasts. 

In accuracy of practical measurements both the stonemasons 
and the irrigation engineers of Egypt of the third millennium 
n.c. reached great heights. It is asserted, for instance, that the 
maximum error in a side and in a corner angle of the Great 
Pyramid are only small fractions of one per cent. Again, the sur- 
veyors responsible for observing the Nile succeeded in placing 
their water gauges in one plane for a distance of about 700 miles 
round all the bends of the river. With a sufficient number of 
centuries for observation, this could be done by trial and error, 
and it docs not necessarily imply any great knowledge of scien- 
tific surveying. The Egyptians had plenty of time. 

In the direction of geometry they seem to have known that 
the area of any triangle is obtainable by the rule •J- base X aid" 



tude. They also computed the volume of a cylindrical granary 
correctly. These results are as advanced as anything the Egyp- 
tians are known definitely to have obtained, with the exception 
of their work on the pyramid to be noted next. For a people who 
achieved magnificent art, it must be admitted that the Egyp- 
tians’ efforts toward geometry are mostly trivial and disappoint- 
ing. This, probably, is only to be expected, as acceptable art is 
created by peoples but little above savagery. 

The greatest Egyptian pyramid 

Every list of the seven wonders of the ancient world includes 
the Great Pyramid. But since the translation of the Moscow 
papyrus 6 in a.d. 1930, this pyramid has been overtopped by a 
greater than any the slaves in Egypt could ever have reared. 
This greatest of Egypt’s pyramids existed only in the mind of a 
nameless mathematician who discovered or guessed the most 
remarkable result in pre-Greek geometry. He gave a numerical 
example of the correct formula, -g-A(fl 2 -f- ab + b 2 ), for the volume 
of the frustum of a truncated square pyramid, h being the 
altitude and a,b the sides of the top and bottom bases. This 
numerical application of a special case of the prismoidal formula 
dates about 1850 n.c. It is not known how this formula was 
obtained. Of several plausible conjectures, none is accepted by a 
majority of reconstructive scholars. 

Had the forgotten Egyptian responsible for this result proved 
his procedure, he would rank high among the greater creators of 
mathematics. Even the empirical discover}’- of such a process or 
its verbal equivalent is evidence of extraordinary mathematical 
insight. In some guise the essential method underlying the 
formula has reappeared in all the great ages of mathematics. 
The Greeks called it exhaustion; 6 Cavalieri in the seventeenth 
century called it the method of indivisibles and, as will appear 
in the proper place, got no closer to proof than the ancient 
Egyptians of at latest 1850 n.c. To us it is the theory of limits 
and, later, the integral calculus. The reasons for believing that no 
Egyptian could ever have even distantly approached a proof 
recur many times in mathematical history; the final and con- 
clusive one was stated only in a.d. 1900. 

The complete method of exhaustion is sufficiently described 
through the simpler problem of determining the area of a circle. 
Regular polygons of n sides arc inscribed and circumscribed to 
the circle; the required area is less than that of the circum- 


scribed polygon and greater than that of the inscribed; as n is 
increased, the difference between the areas of the polygons 
diminishes until, in the limit, as n tends to infinity, the differ- 
ence vanishes, or is ‘exhausted,’ and the common area of the 
limiting polygons is equal to that of the circle. In many partial 
applications of the method, only inscribed polygons were con- 
sidered. In either variant, it is necessary to know the area of a 
regular polygon of n sides. This is immediate once the area of an 
isosceles triangle is known. If the limits described exist, and if 
they can be calculated, the problem is solved. 

At any stage, say n = 96, where Archimedes stopped in the 
third century b.c., an approximation to the area of the circle is 
obtained from the calculable polygons. Moreover, this approxi- 
mation is comprised between determinate bounds given by the 
areas of the inscribed and circumscribed polygons of 96 sides. 
But the crucial step in obtaining the exact formula for the area, 
or even defining what is meant by the area , is taken only by passage 
to the limit as n becomes indefinitely great. 

For the truncated square pyramid we might proceed similarly 
by inscribing and circumscribing stairways whose steps are rec- 
tangular prisms with square bases; and it is conceivable that the 
Egyptian inferred his rule from the easily calculated approxima- 
tions given by stairways with a few steps. Indeed, the earlier 
pyramids were of this type, and the Great Pyramid itself pre- 
sented just such an appearance before the final smooth sheathing 
of dressed stone was applied. But however the Egyptian reached 
his rule, his intuition gave him the correct result that is provable 
only by the integral calculus in some guise. For all proofs of the 
prismoidal formula and its special cases ultimately appeal to the 
formula for the volume of a triangular pyramid. The trivial 
generalization of this result for a pyramid on any polygonal base 
was attributed by Archimedes to Democritus, the founder of 
atomism, in the fifth century b.c. Criticism by the Greek sophists 
of the limiting processes used by Democritus and others was 
partly responsible for the particular course which mathematics 
followed in ancient Greece; and this was one of the major turning 
points in the evolution of mathematics. It would be interesting 
to know whether Democritus was influenced by the Egyptian 
result. He was one of the most widely traveled and more boastful 
of the early Greeks, bragging that although the Egyptian rope 
stretchers taught him all they knew, he himself knew far more. 

Might it not be possible, it may be asked, that the Egyptian 



obtained his rule by some device obviating the theorem of 
Democritus ? This would be so only if it were possible to prove 
by elementary means that triangular pyramids of equal altitudes 
are to one another as their bases. Euclid’s proof in his Elements 
is not elementary in that it is by the method of exhaustion, 
implying the concept of continuity. It was this concept to which 
the sophists objected so pertinently that they succeeded in de- 
flecting mathematics into a narrow channel which, to a modern 
mathematician, seems forced and unnatural. The consideration 
of this will be our principal concern when we follow mathematical 
thought through ancient Greece. 

Conceivably, a strictly finite proof of Euclid’s basic theorem 
for triangular pyramids might be possible. The lack of such a 
proof might be due only to mathematical incapacity and not to 
the nature of mathematics. Were such a proof possible, the 
Egyptian might well have proved his rule, or at least have 
perceived, however dimly, some mathematical grounds for it. 

Recognizing the fundamental mathematical significance of 
the possibility of a strictly finite proof for Euclid’s theorem, C. 
F. Gauss (a.d. 1777-1855, German), usually rated with Archi- 
medes (287 ?— 212 b.c., Greek) and Isaac Newton (a.d. 1642- 
1727, English) as one of the three greatest mathematicians in 
history, in 1844 urged that a proof not depending upon con- 
tinuity be sought. Thus it was by no means obvious to Gauss 
that such a proof might not be found. In 1900, M. W. Dehn 
proved that no such proof is possible. 

It seems unlikely, then, that the Egyptian had anything 
resembling a proof for his rule. If it was just a lucky guess, he was 
so good at guessing that he needed no mathematics. Several of 
the greater mathematicians have emphasized intuition in mathe- 
matics as the necessary spark without which there is no 
discovery. Some have even discounted proof almost to zero, 
claiming that any competent hack can grind out a proof once 
the result has been guessed. Measured by this demotic standard, 
the nameless Egyptian was a very great mathematician indeed. 

The contribution of Babylon and Egypt 

It has been said that no subject loses more when divorced 
from its history than mathematics. This may be true, but there 
is a sort of converse which is equally true. The history of no sub- 
ject loses more when divorced from its subject than does the 


history of mathematics. With this in mind, we recall that we are 
primarily interested in the development of mathematical 
thought, rather than in the exhibits in a museum of antiquities. 
It is now time to apply this primary interest to our first collection 
of specimens. With the memorable achievements of Babylon 
and Egypt behind us, let us glance back for a moment, forget all 
the human struggling that made these long-dead things once 
live, and estimate them solely in the light of mathematics. One 
item of all the treasure will suffice as typical of all. The Babylo- 
nian mensuration of the circle throws into sharp relief the distinc- 
tion between what is mathematics and what merely resembles 

In the familiar formulas lirr, 7 rr 2 for the circumference and 
area of a circle of radius r, 7 r denotes a constant number which, to 
seven decimals, is 3.1415926. The last of course is of great 
practical importance. But the long chronicle of 7r signifies vastly 
more for the history of mathematics than a rather dreary record 
of successive approximations from the crude 3 of the Babylo- 
nians of about 2000 Rtc. to the 707 decimal places, all but a few 
of them quite useless, of W. Shanks in a.d. 1853. 

Any tyro in geometry understands what is intended by such 
an elliptical statement as “the ancient Babylonians took7T equal 
to 3.” But accepted literally, this statement implicitly denies the 
existence of mathematics and makes nonsense of its history. So 
far as is known, nobody before the ancient Greek mathemati- 
cians ever “took7T equal to” anything. Until it had been proved 
that the ratio of the circumference of any circle to its radius is in- 
dependent of the radius, or that (Euclid, XII, 2) the areas of any 
two circles are to one another as the squares on their diameters, 
there was no “ 7 r” to be taken. 

Induction from physically measured circles may have sug- 
gested to some empiricist that the circumference of any circle 
is greater than 3\\ diameters and less than 34, the bounds 
proved by Archimedes to exist. But only a mind very immature 
scientifically would trust these bounds for circles either so small 
or so large that they could not be measured by the means used 
for the others. Certainly no mind with the faintest stirrings of a 
mathematical instinct would trust them. Induction from practi- 
cal experience is not enough here; mathematics is demanded. 

If, in this particular matter of t, it be argued that the 
ancients before Greece had no need for mathematical — not 
merely numerical — precision, and that close induction from 



experience sufficed, several replies may be given, all pertinent 
and applicable to the entire history of mathematics. First, on the 
severely practical side, any civilized people using a calendar 
would need sooner or later to know, or at least to believe, that 
there is a constant, say c, such that the circumference of a circle 
is c times its radius. Otherwise, as their astronomy became more 
exact, they would live in constant dread that their calendar 
might begin fluctuating disastrously, and with it, their commerce 
and agriculture. 

Second, mathematical precision and numerical precision are 
very different things, in spite of what some practical souls may 
imagine to the contrary. A fair degree of numerical precision was 
demanded in ancient times. Had civilization crystallized in the 
second millennium before our era, no greater precision in nu- 
merical calculations than that which sufficed in Babylon would 
have been required. But, to cite only three instances of the need 
for greater numerical precision as civilization evolved, the 
calendar, geography, and navigation demanded an increasingly 
precise astronomy, and this was forthcoming only when arith- 
metic and geometry had progressed far beyond the sharpest 
exactness possible to mathematical empiricism. The validation 
of a formula is its proof, without which precision even in the 
narrowest sense of practical utility is impossible once the earlier 
stages of civilization are passed. 

A third distinction which sharply separates the Archimedean 
mensuration of the circle from the Babylonian is exactly the dis- 
tinction between scientific and prescientific thinking. A mind 
which rests content with a collection of facts is no scientific mind. 
The formulas in a mathematical handbook are no more mathe- 
matics than arc the words in a dictionary a literary masterpiece. 
Until some unifying principle is conceived by which an amor- 
phous mass of details can be given structure, neither science 
nor mathematics has begun. 

The first and most extensive of all the structures unifying 
number and form is deductive reasoning. There is no conclusive 
evidence that such reasoning was used in mathematics before 
the Greeks. They also advanced far beyond mythology in thcii 
attempts to unify r their observations of nature. Their cosmic 
speculations may have been too naive to be of much scientific 
value; nevertheless, they were deliberate steps away from 
mythology and superstition and toward science. With the con- 
scious recognition that unity and generality are desirable values 


both practically and aesthetically, mathematics and science 
became possible. 

All this may sound rather dogmatic, but it is not so intended. 
It is merely one of two possible points of view; and the reader 
is recommended to take the opposite side, follow it out consist- 
ently, and observe to what conclusions he is led in his estimates 
of mathematics and its history. The like applies to our entire 
future course, and in particular to the following paragraph, with 
which no doubt many will disagree. It is the estimate 7 of pre- 
Greek mathematics to which our argument has led us. 

Until, if ever, evidence is uncovered proving that the Greeks 
were anticipated in their conception of mathematics as a deduc- 
tive science, the greatest contribution of the Babylonians and 
Egyptians must remain their unconscious part in helping to 
make possible the golden ages of Eudoxus and Archimedes. It 
was enough, and should preserve their memory as long as mathe- 
matics lasts. 


For their percussions on the higher arithmetic of Fermat and 
others in the eighteenth to the twentieth centuries, the figurate 
numbers of the Pythagoreans (sixth to fifth centuries b.c.) 
may be remembered as one of the most suggestive contributions 
of arithmetica to the modern higher arithmetic. These numbers 
also achieved a certain prestige in Plato’s science, as for example 
in his Timaeus. The triangular numbers in particular, when 
insinuated into the Empedoclean chemistry of the four 
‘elements,’ earth, air, fire, and water, were partly responsible 
for the remarkable metaphysical conclusion that “all matter 
is essentially triangles.” The figurate numbers are supposed to 
have originated from the representation of the regular polygons 
by placing a pebble on each vertex and then bordering the 
polygons in such a manner that regularity and number of sides 
were preserved. This possible origin has been instanced as an 
early occurrence of the connections between number and space. 
Whether this superficial connection is more than a mathe- 
matical pun, the square numbers pebbled out as described may 
be responsible for the persistence of ‘squares’ in our algebra, 
where geometric imagery is not only obsolescent but irrelevant. 

Another numerical item that might be credited to arith- 
metica as practiced by the Pythagoreans is the law of musical 
intervals, traditionally attributed to Pythagoras himself. The 
law relates the pitches of notes emitted by plucked strings 
of the same kind, under equal tensions, to the lengths of the 
strings. This discovery, the first in mathematical physics, 
revealed an unexpected interdependence of number, space, and 
harmony. It is scarcely surprising, then, that it precipitated a 
deluge of number mysticism. Human credulity being what it 
is, the resulting crop of esoteric philosophies and bizarre creeds 
which sprang up in ancient times, and which continue to flourish 
in our own, might have been anticipated. The Pythagorean law 
was also responsible for the retention of ‘music’ in the standard 
medieval curriculum. In fact, nearly every conceivable use, 
except a sensible one from a modern point of view, was made of 
the epochal discovery that musical sounds and numbers are 
related. The fact that had been discovered by experiment 
became the occasion for abandoning experiment in favor of the 
unaided human reason. Consequently, the experiment that 
might have started a scientific age, in the modern sense, aided 
most effectively in retarding that age for about 2,000 years. 

In logistic — computation — the Greeks did nothing that is 



not best forgotten as quickly as possible by a mathematician. 
Their best attempt to symbolize numbers was a chiidish scheme 
little better than juxtaposition of the initial letters of number 
names. Yet the development of Greek numeration, such as it 
v.'as, might legitimately merit a great expenditure of erudition, 
time, and space in the antiquarian history of mathematics. Its 
interest here is negligible because, fortunately for mathematics, 
Greek numeration quickly perished. Only one of its many dis- 
abilities was its incapacity to represent even moderately large 
numbers concisely. Archimedes in the third century b.c. over- 
came this in a scheme of counting by eighth powers of ten. But 
as he just missed the place-system of numeration, his ingenious 
idea also perished. 

It is supposed that the Greeks themselves, except a few 
experts, made little use, if any, of their alphabetic numbers in 
computation, but resorted to the abacus. Sporadic attempts to 
rehabilitate the battered reputation of Greek logistic as a work- 
able system appear to be based on misapprehensions of what 
the Greeks actually did; and the majority opinion remains that 
of the conservative and sympathetic historian of Greek mathe- 
matics who characterized Greek numeration as vile. Some good 
thing undoubtedly came out of Nazareth, but it seems unlikely 
that any decent arithmetic could have issued from an inherently 
vile way of symbolizing numbers, and those who have gone most 
deeply into the matter assert that none did. 


Computation begins only after the mathematics has been 
done. Neither the Hindu numerals nor any others are of any 
importance whatever in vast tracts of modern mathematics. No 
numerical computations are performed. 

Gauss is reported to have lamented that his ancient compeer 
Archimedes just failed to anticipate the Hindu system of nu- 
meration. With his own prodigious astronomical calculations 
behind him, Gauss speculated how much farther advanced the 
science of the nineteenth century might have been had Archi- 
medes succeeded. If this report is accurate, it admirably points 
the parting of the ways. For Gauss had computational astronomy 
in mind; and it was Gauss the expert calculator, not Gauss the 
creative mathematician, who was lamenting the failure of 
Archimedes to take that last simple but essential step. 

Ex Oriente lux 

In an older day the sudden rise to maturity of Greek mathe- 
matics was classed with the miracles. Before the twentieth-cen- 
tury research on the records of Babylon and Egypt, it appeared 
that mathematics in Greece had grown from conception to 
vigorous manhood in a mere flash of about three centuries. 
Today we know that the respect which Greek writers themselves 
expressed for the wisdom of the East, even while extolling their 
own, was justified. The sudden maturity is no longer incredible. 
Modern science, beginning with Galileo and Newton, has 
developed with equal rapidity from origins relatively no more 
promising than those from which Greek mathematics evolved. 

The route by which the learning of the East reached Greece 
has yet to be uncovered in detail. But the battles of Marathon 
(490 b.c.), Thermopylae, and Salamis (480 b.c.), where the 
Greeks broke the Persians on land and sea, may have been a 
turning point in mathematical history as they were in that of 
all Western civilization. Those battles prove at least that young 
Greece was in close contact with ancient Persia, the imperial 
successor of Egypt, Babylonia, Phoenicia, Syria, and all Asia 

Marathon and Salamis are universally acclaimed by partisan 
humanists as unmixed benefits for the development of civilized 
culture. The career of mathematics hints that they may have 
been the beginning of a long detour round the origins of much 
that today is more vital than some of the Greek masterpieces. 

Until the route from East to West is traced, we shall not 


know definitely how much Greek mathematics owed to its prede- 
cessors. Without in any way belittling the Greek contribution, 
we may safely believe that the extreme miracle of spontaneous 
generation did not happen in Greece. In this connection, two 
opposing theories of the anthropologists may be mentioned. Ac- 
cording to the first, closely similar cultures will evolve spon- 
taneously in similar environments, no matter how -widely 
separated. According to the second, all civilization is propagated 
from foci of culture, which in turn were civilized from more 
remote foci, and so on, until all culture is traced to one initial 
focus, usually Egypt. 

A third theory combines the patent advantages of both. The 
spontaneous theory is echoed in the frequent remark that mathe- 
matical and scientific discoveries are often made independently 
and almost simultaneously by two or three men. On the diffusion 
theory this can be explained by observing — -what is the fact — 
that in such instances the discoveries usually germinate in a body 
of knowledge accessible to all. Something roughly like this is now 
believed to have been responsible for the sudden efflorescence 
of Greek mathematics. 

The lore of the East was available to any curious Greek who 
could afford a journey to Egypt and Babylonia. Trusting Greek 
tradition, we may assert that many early Greeks, urged by their 
notorious and childlike curiosity, traveled extensively in the 
East and profited enormously by their travels. Greek mathe- 
matics is sufficient evidence of the insatiable hunger of the 
awakening Greek mind for exact knowledge, and the most 
adequate measure of its intellectual capacity. 

Txuo supreme achievements 

All the immaterial riches of the generous East were anyone’s 
for the asking and the taking. The early Greeks appear to have 
asked for everything and to have taken nearly all, fools’ gold 
along with the rest. In their youthful eagerness to acquire, they 
overlooked two obvious opportunities, each of the first impor- 
tance for the futures of science, mathematics, and philosophy 
then possible. 

The sixth century before Christ was the time, and Greece the 
place, for human beings to reject once for all the pernicious num- 
ber mysticism of the East. Instead, Pythagoras and his fol- 
lowers eagerly accepted it all as the celestial revelation of a 
higher mathematical harmony. Adding vast masses of sheer 


numerological nonsense of their own to an already enormous 
bulk, they transmitted this ancient superstition to the golden age 
of Greek thought, which passed it on in the first century a.d. 
to the decadent arithmologist Nicomachus. He, enriching his 
already opulent legacy with a wealth of original rubbish, left 
it to be sifted by the Roman Boethius, the dim mathematical 
light of the Middle Ages, thereby darkening the mind of Chris- 
tian Europe with the venerated nonsense, and encouraging the 
gematria of the Talmudists to flourish like a weed . 4 

It is customary in the histories of science and mathematics 
to ignore these vagaries of the human mind. However, it would 
seem to impartial onlookers that only a distorted image of the 
not too flattering facts can result from any historical account 
which reports only what are now considered successes and ig- 
nores the failures. Frequently the sense of one epoch has become 
the nonsense of a later, and what no longer is meaningful may 
once have been of prime scientific or social importance. A case in 
point is the phlogiston theory of heat; another, the tripartite 
infinity of theology, a lineal descendant of the mystical arithme- 
tic of the Pythagoreans; and yet another, the Platonic theory of 
mathematical truths, long since abandoned by unmystical 
mathematicians. Without some attention to such misadventures 
in ideas, the development of mathematics, no less than that of 
science, appears as an uninterrupted parade of triumphs with 
never a recession to relieve the glorious monotony. Gratifying 
as such a presentation may be, it is not therefore necessarily 

Had the Pythagoreans rejected the number mysticism of the 
East when they had the opportunity, Plato’s notorious number , 5 
Aristotle’s rare excursions into number magic, the puerilities of 
medieval and modern numerology, and other equally futile 
divagations of pseudo mathematics would probably not have 
survived to this day to plague speculative scientists and be- 
wildered philosophers. Nor would a mathematical astronomer 0 
of the early twentieth century have beheld the astounding 
spectacle of God masquerading as a mathematician. Among the 
gains accruing from the ancient numerology is the inspiration 
for much of Plato’s theory of Eternal Ideas. 

If on the other hand the early Greeks had accepted and 
understood Babylonian algebra, the time-scale of mathematical 
development might well have been compressed by more than a 
thousand years. But to a people just starting to grow up mathe- 



matically, the attractions of a mystical, all-embracing philos- 
ophy were naturally more seductive than those of an austere 

A greater disaster at the height of the Greek golden age held 
mathematics and science back immeasurably. Instead of follow- 
ing the bold lead of Archimedes and developing a fluent, dynamic 
mathematics applicable to the ceaseless flux of nature, the lesser 
Greek mathematicians of the third century b.c., and after, 
lingered behind with the Piatonists and cast their thought in 
geometric shapes as perfect and as rigidly static as the Parthe- 
non. In the entire history of Greek mathematics, all but the 
incomparable Archimedes and a few of the more heterodox 
sophists appear to have hated or feared the mathematical 
infinite. Analysis was thwarted when it might have prospered. 

Such are the debits in the account of Greek mathematics with 
time. They are heavy enough; but beside the credits, they are of 
little moment. It has fallen to the lot of but one people, the 
ancient Greeks, to endow human thought with two outlooks on 
the universe neither of which has blurred appreciably in more 
than two thousand years. From all the mass of their great 
achievement, these two, each of superlative excellence, may be 
exhibited here by themselves, not to diminish their magnitude 
by a crowd of lesser masterpieces, all great but not the greatest. 

The first was the explicit recognition that proof by deductive 
reasoning offers a foundation for the structures of number and 

The second was the daring conjecture that nature can be 
understood by human beings through mathematics, and that 
mathematics is the language most adequate for idealizing the 
complexity of nature into apprehensible simplicity. 

Both are attributed by persistent Greek tradition to Pythag- 
oras in the sixth century before Christ. No contemporary 
record of these epochal advances survives; and there is an equally 
persistent tradition that it was Thales in the sixth century B.c. 
who first proved a theorem in geometry. But there seems to be no 
claim that Thales, earliest of the ‘seven wise men of Greece,’ 
proposed the inerrant tactic of definitions, postulates, deductive 
proof, theorem as a universal method in mathematics. Again, in 
attributing any specific advance to Pythagoras himself, it must 
be remembered that the Pythagorean brotherhood was one of 
the world’s earliest unpriestly cooperative scientific societies, if 
not the first, and that its members assigned the common work of 


all by mutual consent to their master. It is sufficient to remember 
that these advances were made as early as 400 b.c. at the latest, 
and that both were Greek. 

Chronology of Greek mathematics 

Before considering in some detail a few items of more than 
antiquarian appeal, we shall give a short prospectus of the lead- 
ing schools of Greek mathematics with their dates, a few key 
names, and brief mention of the principal advances made by 
each. Some of these will not be noted again. All dates except 
those attached to men’s names are only approximate; those 
•without a.d. are b.c. 

The birth, maturity, and senescence of Greek mathematics 
cover about ten centuries, roughly from 600 b.c. to a.d. 400. 
The earliest period, 640-550, was that of Thales (624?-550?), of 
the Ionian school, and Pythagoras (569?-500?). Its outstanding 
achievements are the founding of mathematics as a deductive 
system and the program of mathematicizing natural phenomena. 

In the fifth century, the Greek sophists of Elea in Italy 
hardly constituted a mathematical school, yet were of funda- 
mental importance for the development of all mathematical 
thought. By his ingenious paradoxes on infinite divisibility, Zeno 
(495 ?— 4-35 ?) cast doubt on some of the reasoning of his predeces- 
sors, and was partly responsible for the characteristically Greek 
course which mathematics entered with the succeeding school 
and followed thereafter. The sophist revolt against plausible 
reasoning thus marks one of the cardinal turning points in the 
history of mathematics. 

The third and fourth schools, Athens and Cyzicus, 420-300, 
are one except geographically. Of the very first importance for 
all the future of mathematics was the disposal of some of the 
sophists’ objections by Eudoxus (408-355), a pupil and at one 
time a friend of Plato, in his theory of proportion. Essentially a 
theory of the real number system, this Greek work of the fourth 
century b.c. was not substantially modified till the latter half 
of the nineteenth century a.d., when critical difficulties in analy- 
sis necessitated a thorough reexamination of the concept of real 

In this period, Plato (429-348) was to assume a mathematical 
importance greatly in excess of any warranted by his own slight 
contributions. The general professional opinion is that Plato s 
too rigid ideal of mathematics 7 as a high philosophic art was to 



cramp and trammel mathematicians abler than himself. How- 
ever, discounting the excessive purity of the Platonic ideal, 
Menaechmus (375 ?— 325 ?), a pupil of Eudoxus and reputedly a 
tutor of Alexander the Great, inaugurated the geometry of conic 
sections. There is a tradition that Plato encouraged Menaech- 
mus. If this is true, Plato made a fundamental contribution to 

A basic technique was added to mathematical reasoning in 
this period by Hippocrates s (of Chios, 470 ?-?; not to be confused 
with the great physician of the same name, of Cos). By exploiting 
it in his own geometry, Hippocrates demonstrated the power of 
the indirect method ( reductio ad absurdum , reduction to an 
absurdity or deduction of a contradiction from an assumed 
hypothesis which it is desired to disprove). The universal validity 
of this method remained unchallenged till the twentieth century, 
when objections were raised to its indiscriminate use in reasoning 
about infinite classes. 

The fifth school was the First Alexandrian, 300-30, in the 
city founded by Alexander the Great in 332. This was the cul- 
mination of Greek mathematics. With the exception of Dio- 
phantus (possibly not a Greek, date conjectured from second 
to fourth century), the rest is anticlimax. In this great age, 
Euclid (365 ?— 275 ?) wove elementary plane and solid synthetic 
geometry into the close deductive system that was to remain the 
school standard for over 2,200 years. He also systematized the 
Greek arithmetica as it existed in his time, and wrote on geo- 
metrical optics. 

In this age lived Archimedes (287-212), the greatest scientific 
and mathematical intellect of the ancient world, also, by virtue 
of the uncramped freedom of his methods, the first modern 
mathematician. Not till England produced Newton in the seven- 
teenth century, and Germany Gauss in the nineteenth, did the 
exact sciences show this ancient Greek his peers. With magnifi- 
cent indifference to the mathematical proprieties of his age, 
Archimedes used whatever came to mind or hand to advance 
mathematics. Unlike many of his fellow Greeks, he did not dis- 
dain experiment. He founded the mathematical sciences of 
statics and hydrostatics. He anticipated the integral calculus, 
also, in the one problem of drawing a tangent to his equiangular 
spiral, the differential calculus. 

In this age also lived the supreme master of the synthetic 
method in geometry. Apollonius (260 ?— 200 ?) left but little for 


his successors in that method to do in the metric geometry of 

Astronomy became a ' mathematical science during this 
period, in the work of Hipparchus (of Rhodes, second half of 
second century b.c.). From Hipparchus on through Ptolemy 
(second century after Christ), Copernicus (fifteenth century), 
Tycho Brahe (sixteenth century), and Kepler (sixteenth century), 
astronomy did not deviate from the Hipparchian program of a 
geometry to describe the motions of the planets. With Newton, 
this geometry evolved in the seventeenth century into dynamics. 
Hipparchus also was the first to use a sort of trigonometry 
systematically, and is said to have produced the equivalent of 
a rudimentary table of sines. 

Geodesy advanced in this period with the measurement, as 
accurate as the available data and instruments permitted, of a 
degree of the earth’s surface by Eratosthenes. This man is re- 
membered also for his reform of the calendar and for his method 
of sifting out the primes from the sequence of all the integers. 

Finally, this rich period produced one of history’s most 
ingenious scientific engineers, Heron (of Alexandria, second cen- 
tury, perhaps not Greek). The formula 

[j(j - a) (s - b)(s - c)]* 

for the area of a triangle with sides a,b,c and 2s s= a -f b + c, 
often attributed to Heron, is of course important in trigonom- 
etry. But its peculiar historical significance is elsewhere. It 
marks a carefree departure from the too rigid niceties of ortho- 
dox Greek mathematics, a departure which was halted all too 
soon. No academic Greek geometer would have presumed to 
‘multiply together four lines,’ as in the formula; for the product 
has no geometrical meaning in Euclid’s space of three dimen- 
sions. The engineer Heron was not deterred by such obstacles. He 
discovered or transmitted the right result and, like the Egyptian 
he may have been, left it for future generations of mathemati- 
cians to show that he had not erred in his own proof. If, however, 
as is now claimed, the formula is due to Archimedes, the mystery 
of it all vanishes. 

The sixth and last school was the Second Alexandrian, 30 
b.c.-a.d. 640. The first date marks the absorption of Egypt by 
Rome, the second 9 the destruction by the Moslems of what little 
Roman virility, Greek neglect, and early Christian intolerance 



had left — some say nothing — of the great library at Alexandria. 
But Greek mathematics had lost most of its creative power long 
before the library disappeared, and only three men in the six 
centuries of the Second Alexandrian school would have been 
noticed as mathematicians by the giants of the First. 

Ancient astronomy culminated in the second century a.d. 
in the eccentrics and epicycles of Ptolemy. For about fourteen 
centuries, Ptolemy’s geocentric description of the solar system 
was to be accepted as ultimate. Geometry and arithmetic had 
long since become independent provinces of mathematics ■when 
Ptolemy, compelled by exigencies of astronomical computation, 
all but split oft trigonometry' as a distinct mathematical science 
in his geometrical theorems equivalent to the addition formulas 
for the sine and cosine, and in his computation of a table of 
chords. The failure of trigonometry to attain its freedom was due 
to Ptolemy’s lack of algebra and the disabilities of logistic. 

Almost at the end of the creative period, a belated geometer, 
Pappus (second half of the third century), either transmitted 
or himself discovered three prophetic theorems. He proved 
the focus-directrix property for the ellipse, parabola, and 
hyperbola, thereby foreshadowing the general equation of the 
second degree for all conics in analytic geometry. He also proved 
in effect that the cross ratio (or anharmonic ratio) of four 
collinear points is a projective invariant, thus isolating a cardinal 
theorem in the projective geometry of the seventeenth and 
nineteenth centuries. Finally, he used one of the numerous dis- 
guises of the integral calculus to obtain the theorem often 
ascribed to P. Guldin (a.d. 1577-1643, Swiss) that the volume 
generated by a plane figure F rotated about a fixed axis is AL , 
A — the area of F, L = the length of the path traced by the 
centroid of F. Obviously it is impossible to give an acceptable 
proof without a full use of the calculus. 

At last, in the rudimentary algebra of Diophantus, also in 
his higher arithmetic, mathematics all but entered a renaissance. 
But it was getting late, and the Greek spirit was too tired to 
return to its point of departure and resume the march begun by 
others some twenty-four centuries before in Babylon. If there is 
such a thing as the Zeitgeist, it must have permitted itself a 
sardonic smile as it prepared the sourest jest in the history of 
mathematics. Not Diophantus, but his historical predecessor of 
the first century after Christ, the numerologist Nicomachus, 
was to transmit arithmetica to Christian Europe. 


Number from Pythagoras to Diophantus 

A few items in the foregoing prospectus overtop the rest 
in importance for what was to be the future of mathematics. 
These will now be examined in closer detail. 

The Pythagorean brotherhood’s conception of mathematics 
was broad and human. All of their philosophy, of which mathe- 
matics was only a subordinate if important part, was directed 
to but the one end of sane, civilized living. Arithmetic, geometry, 
astronomy, and music were the four divisions of their mathe- 
matics. This tetrad was to survive for centuries, passing through 
the Middle Ages in the attenuated quadrivium which formed 
four-sevenths of a liberal education, the rest being the trivium 
of grammar, rhetoric, and logic. By then, however, the Pytha- 
gorean liberality of spirit had been stifled, and living often was 
neither sane nor civilized in any sense that Pythagoras would 
have recognized. 

Of Pythagoras himself only legends remain. In middle life he 
migrated from his native Samos to Crotona in southern Italy, 
where the best work of his brotherhood was done. For the rest, 
fable makes him an enthusiastic if somewhat pompous mysta- 
gogue who had traveled extensively in the East, and who used 
his mystical lore to impress everybody from blacksmiths to 
young women. In liberality of mind he was centuries ahead of his 
time. As but one indication of many that the Pythagoreans 
sought to enlighten their contemporaries, women were admitted 
to the master’s lectures; and Pythagoras himself seems to have 
had no use for the very peculiar masculinity of the Athens of 
Socrates and Plato. It is said that Pythagoras and his immediate 
disciples perished in the flames kindled by those whom they had 
striven to deliver from brute ignorance, prejudice, and bigotry. 
In any event, the Pythagoreans were driven out to seed their 
wisdom elsewhere. 

The evil that accrued from the Pythagorean numerology has 
been sufficiently noted. But some good also issued from it at 
a very long last. From nonsensical hypotheses the Pythagoreans 
deduced that both the sun and the moon shine by light reflected 
from a Central Fire. Some twenty centuries later, Copernicus 
(or his officious editor) in his dedicatory epistle to the reigning 
Pope stated that this wild deduction gave him a hint for his 
own heliocentric theory of the solar system. 

Again, readily discovered relations between the positive 



integers of certain categories, such as the odd or even, or the 
polygonal numbers, might easily delude an imaginative mind 
into ascribing human and superhuman powers to number. When 
Pythagoras discovered the ratios -f, ■§, -jj for the lengths of 
plucked strings under the same tension to give the octave, the 
fifth, and the fourth of a note, the first recorded fact in mathe- 
matical physics, it was an understandable extrapolation that 
“Number rules the Universe,” and that the ‘essence’ of ail things 
is number. In the pardonable enthusiasm of that too-inclusive 
generalization, the modern theory of the continuum of real 
numbers originated. 

To follow the clue from Pythagoras to the present, we must 
return to Thales. He too had learned much from the wise men of 
the East. The story that he predicted a solar eclipse in 585 b.c. 
appears to be apocryphal. 10 The like may be true for the equally 
famous and mathematically more important legend 10 that, while 
in Egypt, Thales estimated the height of the Great Pyramid by 
an obvious application of similar triangles to the shadow of the 
pyramid and that cast by his staff when held perpendicular to 
the ground. 

Whether or not the legend records a fact, the Pythagoreans 
by the fifth century b.c. had reached a critical stage in the 
development of the number concept. For they proceeded to 
prove that if a,b,c and a',b',c' are corresponding sides of similar 
triangles, then a/b — a'/b', b/c — b'/c'. (The remarks on history 
and proof in the Prospectus arc particularly relevant here.) 

By the fourth century b.c. it was perceived that the Pythago- 
rean proof concealed the subtle assumption that the numbers 
measuring the sides a,b,c,a',b\c r are rational, that is, each is 
expressible as the ratio (quotient) of two integers. But these 
sides had been assumed to be of any finite lengths whatever. 
Hence it had been assumed that there is a one-one correspond- 
ence between the lengths of straight-line segments and the 
rational numbers. In particular, it had been assumed that the 
length of the diagonal of a square whose side is a rational number 
is itself a rational number. If the side is 1 unit, the diagonal is 
VI units in length. But the Pythagoreans easily proved that VI 
is not expressible in the form m/n, where m, n are integers. 

It would be of great interest to know who 11 first proved the 
irrationality of VI» but probably we never shall. In reply to a 
question by Socrates, Thcaetetus says, “Thcodorus was writing 
out for us something about roots, such as the roots of three or 


five feet, showing that in linear measurement (that is, comparing 
the sides of the squares) they are incommensurable by the 
unit; he selected the numbers which are roots, up to seventeen, 
but he went no farther; and as there are innumerable roots, the 
notion occurred to us of attempting to include them all under 
one name or class.” But this does not settle the vexed question of 
who first proved the irrationality of V2; and anyone who wishes 
may still believe without danger of contradiction by incon- 
trovertible evidence that Pythagoras himself did. All we need 
keep in mind is that the Pythagoreans by the end of the fifth 
century b.c. knew that V 2 is irrational. With great ingenuity 
they approximated to by successive solutions of the equa- 
tions 2a; 2 — y 2 = +1, 

Two ways of proceeding lay open. The choice was between 
some lengths corresponding to no number, or and other 
positive irrationals being numbers. Choosing the second, the 
geometers of the fourth century b.c. passed one of the epochal 
milestones in the history of all thought. “The grand continuum” 
of analysis, the real number system, was already in view. So 
also were the paradoxes of the infinite. Unless the new numbers, 
the positive irrationals, could be incorporated with the positive 
rationals in a unified domain of ‘numbers’ or ‘magnitudes’ so 
that all should form a self-consistent system under the operations 
addition, subtraction, multiplication, and division as then under- 
stood for the rationals, the newly imagined irrationals would be 
illusory. Further, operations in the enlarged number system 
must yield the same results for the rational numbers as before 
the adjunction of irrationals to the rational number system. 
The demand for internal consistency in the enlarged domain was 
automatically imposed, for it had already been agreed that 
mathematics should not defy strict deduction. 

No further extension was made until the seventeenth century, 
when the negative numbers were fully incorporated (but with- 
out mathematical understanding) into the real number system. 
About 1800 the final step was taken when the imaginaries 
were adjoined to the completed real n umbe r system, and 
the domain of complex numbers (a + b \ — 1, a, b real) was 
created. In both of these later extensions, the underlying method- 
ology of generalization and internal consistency had not changed 
since the fourth century b.c. The Greeks appear to have been 
guided by subconscious mathematical tact. Explicit formulation 



and perhaps clear understanding of the methodology of extend- 
ing the number system came only in the late nineteenth century. 
We shall return to this presently. 

Thus far only half the project of adjoining the irrationals, 
as the Greeks saw the problem, had been imagined. They thought 
in spacial imagery and had generalized their conception of 
geometrical ‘magnitude’ to include both rational and irrational 
magnitudes. It did not occur to them immediately that the more 
difficult half of the project remained undone. They had still to 
prove that their enlarged system of magnitudes was self-con- 
sistent; and they appear either to have overlooked this necessity 
entirely at first, or to have considered it obviously satisfied. 
Looking critically at what appeared obvious — one of the almost 
infallible ways of making a fundamental addition to mathemat- 
ics — they discovered that it was not all obvious. On close inspec- 
tion they perceived difficulties that have not been completely 
resolved even today. As already emphasized, the manner in 
which Eudoxus surmounted these difficulties marks a major 
turning point in the long history of mathematics. 

It was impossible for the Greeks or anyone else to understand 
cither geometry or the real number system without some theory 
of continuity in the mathematical sense. This incidentally neces- 
sitated a clarification of the limiting processes, such as exhaus- 
tion, already described in connection with the Egyptian 
mensuration of the pyramid. Until such processes were strictly 
validated it was nonsense to speak of the area of a circle, or of the 
volume of any solid, or of the length of any line, straight or 
curved, except only when the numerical measures of such areas, 
volumes, and lengths were rational numbers. As irrational 
measures arc infinitely more numerous (to the power of the 
continuum) than the rational, mensuration and the geometrical 
theory of proportion scarcely existed before Eudoxus. 

The necessity for drastic revision was strikingly emphasized 
by Zeno in four ingenious paradoxes or, as some might say, 
sophistries. A sophistry in mathematics is a logical argument 
that some dislike but cannot refute. Zeno’s classic four have 
probably occasioned more inconclusive disputation than any 
equal amount of disguised mathematics in history. 

Zeno’s service to mathematics is so outstanding that it would 
be interesting to know something of the man himself. Very little 
has survived. By tradition, he was a pugnacious dialectician 
with a passion for being different from everyone else. In middle 


age he was “of a noble figure and fair aspect.” His paradoses 
are evidence enough of an independent mind, and it is told that 
his uncompromising intellectual honesty finally cost him his life. 
He had conspired with the political faction which lost, and met 
his death by torture with heroic fortitude. The first of his para- 
doses will suffice here. 

Zeno argued that you cannot get to the end of a racecourse, 
because you must traverse half of any given distance before you 
traverse the whole, and half of that again before you can traverse 
it, and so on, ad infinitum. Hence there are an infinite number of 
points in any given line, and “you cannot touch an infinite 
number one by one in a finite time.” Therefore you will never get 
to the end this side of eternity. 

In this and another of the same type (Achilles and the 
tortoise), Zeno argued against the infinite divisibility of space 
and time. To show his philosophic impartiality, he devised two 
equally exasperating paradoxes on the other side: if finite spaces 
and times contain only a finite number of points and instants, 
we again deduce consequences contradicted by experience. 

Mathematics had freely used the concept of infinite divisi- 
bility. Thus Zeno’s paradoxes, in addition to affording grounds 
“for almost all theories of space and time and infinity which 
have been constructed from his day to our own,” showed that 
geometry and mensuration in the fifth century b.c. needed a new 
foundation. Eudoxus provided this in his theory of proportion, 
applicable to any real ‘magnitudes.’ 

Before recalling how Eudoxus met this ancient crisis, we note 
the nineteenth-century way out of Zeno’s difficulties. By a sim- 
ple application of infinite series it is easily shown that the 
runner will reach his goal and that Achilles will pass the tortoise. 
But — a reservation of the first importance — the logic of con- 
tinuity supporting the modern theory of convergence descended 
in the twentieth century to a deeper level than any that had been 
explored when the nineteenth-century analysts imagined they 
had disposed of Zeno’s paradoxes. 

Eudoxus based his theory on his definition of ‘same ratio:’ 
The ratio P/Q is said to be the same as the ratio X/Y , when, 
m and n being any (positive) integers whatever, mX is greater 
than, equal to, or less than nY according as mP is greater than, 
equal to, or less than, nQ. If the ratios P/Q, X/Y are the same, 
P, Q , X, Y are called proportionals. The theory is expounded in 


Euclid’s Elements, Book V. Book VI contains the application 
to similar figures. 

Some modern critics, particularly among the French, have 
been unable to appreciate the radical distinction between this 
Greek theory of the real number system and that now current, 
due to J. W. R. Dedekind (1831-1916, German). Beginning in 
the 1870’s, the criticisms continued well into the twentieth 
century, especially during the nationalistic fervors generated by 
the first world war. The later critics appeared to be unaware 
that Dedekind had disposed of their contentions in 1876. The 
point of historical interest is that no current theory of real 
numbers is that which sufficed for the fourth century b.c. 

The Eudoxian theory of proportion indirectly validated the 
empirical rule of the Egyptians for the volume of a truncated 
pyramid, and completed the work of the Pythagoreans on similar 
figures. It also certified the method of exhaustion and, after 
Dedekind (1872), the use of the integral calculus in the deter- 
mination of lengths, areas, and volumes. In short, it provided 
a foundation for the real number system of mathematical 

It may be significant that Eudoxus was another of the great 
Greek mathematicians who is said to have visited the East. At 
one time a protege and friend of Plato, he left Plato’s Athenian 
Academy to found his own school at Cyzicus when Plato — so it is 
said — began to show signs of most unphilosophical envy and 
jealousy. But this is hardly credible of the man who composed 
the Lysis and the Symposium. 

The italicized phrase in the definition of ‘same ratio’ illus- 
trates the fact that finality is as hard to reach in mathematics 
as it is in philosophy. For not all schools of mathematical 
thought in the twentieth century have admitted ‘any integers 
whatever’ as a legitimate concept in deductive reasoning. The 
phrase conceals an infinity of trials on all the integers m, n to test 
the inequalities mX | nY, mP | nQ. Thus a consisent finitist, 
if there is one, might say that Eudoxus produced a milder para- 
dox of Zeno’s racecourse; for “you cannot test for an infinite 
number of pairs of integers in a finite time.” However, the 
influential mathematical schools ignore such sophistries, and 
continue to create new mathematics of great interest and 
indubitable scientific utility. 

Having passed this outstanding landmark in the develop- 


ment of mathematical thought, we shall look forward from it for 
a moment before proceeding to the next stage in the Greek 
elaboration of number. Its specific importance in relation to 
geometry, mensuration, and the real number system has been 
sufficiently indicated; and anyone with the slightest feeling for 
great mathematics will admit its greatness without any reserva- 
tion. Contemplating this masterpiece, mathematicians may be 
pardoned a little pride that it was their guild which fashioned it. 
But to leave the Greek masterpiece in sculptured isolation as a 
monument for all time to the perspicacity of the mathematical 
intellect would be to give a totally false impression of the manner 
in which mathematics has developed. Its history is not the record 
of one brilliant victory after another. Rather is it a somewhat 
sobering chronicle of intelligence fighting desperately against 
tremendous odds to overcome the all but ineluctable stupidity of 
the human mind. That such progress as has been made should 
have been possible at all is the miracle of the ages. 

With the detailed example of the Greek attack on incommen- 
surables (irrationals) set out before them with all the elaborate 
precision of a copybook for children, mathematicians stumbled 
about for twenty centuries before imitating the Greek method- 
ology in their struggle to incorporate negative and complex 
numbers with the positive real numbers in a single self-consistent 
system. The irrationals first appeared in geometry, the negatives 
in arithmetic and algebra, and the imaginaries in algebra. Both 
negatives and imaginaries entered when it was gratuitously 
assumed that certain rules of operation, known to produce 
consistent results in special circumstances, would retain their 
validity in all circumstances of a superficially similar kind. In 
Greek geometry, the rules in question were those used in proofs 
concerning similar triangles with rational sides; the tacit assump- 
tion was that these same rules would give consistent results for 
all triangles. In algebra, the negatives and imaginaries entered in 
an analogous way with the solution of equations; and just as the 
Pythagoreans were reluctant to grant the status of number to 
the irrationals, so were the earlier algebraists unwilling to admit 
negatives and imaginaries as legitimate roots of algebraic 

The Greeks recognized that they were confronted by a funda- 
mental problem, isolated it, and solved it. Perhaps the decisive 
step was their bold hypothesis that a ‘magnitude’ (number 
represented geometrically) need not be rational in order to be a 



‘magnitude.’ They generalized the concept of magnitude as it 
first presented itself to experience and intuition. The algebraists 
up to the seventeenth century failed to recognize that the nega- 
tives and imaginaries presented a problem at all. They either 
blindly manipulated such things when the rules for solving 
equations turned them out, or rejected them without attempting 
to justify the rejection. A contemporary of Eudoxus would have 
wanted to know why some equations produced only intelligible 
roots, others only some intelligible, and others none. In their 
lack of common mathematical curiosity, the algebraists of Islam 
and the European Renaissance were contemporaries of the 
ancient Egyptians. They wondered and were perplexed, of 
course; but there they stopped, because they lacked the Greek 
instinct for logical completeness and generality. 

It was only in the nineteenth and twentieth centuries that 
these difficulties ivere satisfactorily met, and then it was by 
a methodology abstractly identical with that of the Greeks. 
First, it was ascertained what algebraists were subconsciously 
striving to do. They were attempting, with no mathematical 
justification, to include all the reals and all the imaginaries in 
one system closed under the four rational operations (addition, 
subtraction, multiplication, division) of common algebra. They 
actually proceeded on the tacit assumption that such closure was 
a mathematical fact, namely, that it was self-consistent. This is 
what they wished it to be, in order to certify their empirical 
calculations. No progress was made until they followed the Greek 
lead in geometry and stated explicit postulates for the real and 
complex numbers, thereby defining the ‘numbers’ presented in 
algebraic experience. Thus the concept of ‘ number’ was extended 
to cover all sets closed under the four rational operations. 
Finally, as will be seen, it was proved in the late nineteenth 
century that the most general set of this kind, in which xy = 0 
only if at least one of x, y — 0, is that of all complex numbers 
a -f b'Sf’-l, a i b real; and that the only such sets are this itself 
and certain of its subsets, for example, the set of all rational 

When wc reflect that it took the Greeks less than two cen- 
turies to recognize and reach their goal, we may well wonder 
whether mathematics today is not starting on another of its 
two-thousand-year quests in search of simplicity. With all its 
prolific inventiveness, mathematics seems to have lost some of 
its youthful directness. Nearly always it is the recondite and 


complicated which is elaborated first; and it is only when some 
relatively unsophisticated mind attacks a problem that its deep 
simplicity is revealed. 

In their further encounters with number, the Greeks found 
much that underlies some living mathematics, but nothing, per- 
haps, of such abiding significance as the work of Eudoxus. Like 
the similar triangles of Thales which were partly responsible 
for the Eudoxian theory, one origin of some of the most interest- 
ing Greek higher arithmetic was in Egypt. 

The Egyptian ‘rope stretchers’ laid out right angles for the 
orientation of buildings by means of a triangle of sides 3,4,5. A 
string of length 3+4 + 5 was marked or knotted at the points 
3,4. With this and three pegs a right-angled triangle was obtained 
in an obvious way. Instead of the particular positive integer 
solution 3,4,5 of a 1 — W + c 2 , they might have used any other, 
provided they knew any. The general positive integer solution 
a, b, c was given by Euclid in his Elements (X, 28, Lemmas). This 
appears to be the first proved complete integer solution of an 
indeterminate equation. Whether first or not, it is the germ of 
vast theories in the modern higher arithmetic and, less directly, 
of the like in algebra. To complete the record, it must be noted 
that since 1923 it has been customary to deny that the Egyp- 
tians ever used 3, 4, 5 to lay out right angles. The argument on 
which this denial is based appears to run as follows. Because the 
rope stretchers stretched their ropes for purposes other than 
laying out right angles, therefore they did not lay out right 
angles by rope stretching. Further, because right angles were 
laid out by other means, therefore they were not laid out by — 
etc. It can be asserted only that the history here may be sounder 
than its supporting logic. 

The solution in integers, or in rational numbers, of indeter- 
minate equations belongs to diophantine analysis. The name 
honors Diophantus, whose treatise of thirteen books, of which 
only six survive, was the first on the subject. The Latin transla- 
tion (a.d. 1621) of this suggestive fragment directly inspired 
Fermat to his creation of the modern higher arithmetic. It also 
inspired something much less desirable. Diophantus contented 
himself with special solutions of his problems; the majority of 
his numerous successors have done likewise, until diophantine 
analysis today is choked by a jungle of trivialities bearing no 
resemblance to cultivated mathematics. It is long past time that 
the standards of Diophantus be forgotten though he himself be 



remembered with becoming reverence. For the opinion on thiB 
matter of an expert in both the history and the practice of 
diophantine analysis, those interested may consult L. E. Dick- 
son’s History of the theory of numbers, vol. 2, 1920. 

On another account also this work of Diophantus is memora- 
ble. It was the first Greek mathematics, if indeed it was Greek, 
to show a genuine talent for algebra. Following the Pythago- 
reans, Euclid had given geometrical equivalents for simple 
identities of the second degree, such as a(a -f b) = a~ -f- ab, 
(a -f b) 2 = a 1 -f- b- -f- 2 ab, and had solved x 1 -{- ax — a-, a 
positive, geometrically. Diophantus gave essentially algebraic 
solutions of special linear equations in two and three unknowns, 
such as x -f- y = 100, x — y — 40. More important, he had 
begun to use symbols operationally. This long stride forward is 
all the more remarkable because his algebraic notation, com- 
pared to that of today or of the seventeenth century when 
Descartes practically perfected it, was almost as awkward as 
Greek logistic. That he accomplished what he did with the avail- 
able technique places him beyond question among the great 

His operational advance was profoundly significant. In 
algebra a formula, say a -f- b — c, directs us to perform certain 
operations on given numbers (or, in modern algebra, abstract 
marks), here an addition and a subtraction on a , b, c , in a pre- 
scribed order. That is, algebra escapes from verbal instructions 
to symbolic directions and ceases to be purely rhetorical. Dio- 
phantus had even invented a species of minus sign, and permitted 
a negative number to function in an equation on a parity with 
positive numbers. He also used symbols for the unknowns and 
for powers. All this was a long step toward symbolic algebra. It 
seems probable that some of Diophantus’ algebra was of Baby- 
lonian origin, although the connection has yet to be traced. 
Unfortunately for the development of algebra and of mathe- 
matics generally, Diophantus v'as at least four centuries later 
than Archimedes. 

To conclude this account of Greek arithmctica, we may 
return to its origin in geometry and instance the timelessness of 
great mathematics by an episode in arithmetic and geometry 
from the twentieth century. 

The Pythagorean theorem that x- + y- = z~, where x,y,z are 
the sides of a right triangle, is the basis of metric geometry in 
Euclidean space. In the spaces defined by Riemann (1854), 


the quadratic algebraic form x 2 + y 2 in two variables, x, y is 
replaced by a quadratic differential form in n variables; n = 4 
is the case of interest in relativity. The significance of x 2 -f y 2 
= z 2 in diophantine analysis has been remarked. There, this 
equation is also generalized, and it is required to solve the 
general quadratic equation in n unknowns, with integer coeffi- 
cients, in integers. This arithmetical problem, together with 
that of reducing the general equations of the second degree in 
the analytic geometry of conics and quadrics to canonical form, 
suggested (nineteenth century) the purely algebraic problem of 
reducing a quadratic form in n variables to a sum of squares 
each multiplied by an appropriate coefficient. Incidentally, this 
problem is important in dynamics. 

Toward the close of the nineteenth century (a.d. 1882), a 
notable advance was made in the diophantine problem by 
Minkowski, then a youth of eighteen. In treating this problem, 
Minkowski acquired a mastery of the algebraic theory of the 
reduction of quadratic forms. Becoming interested about the 
turn of the century in mathematical electromagnetism, he 
applied his algebraic skill to special differential forms. By the 
peculiar accident of his interests, he was the ideal candidate to 
recast the mathematics of Einstein’s special relativity of a.d. 
1905 into a shape which still retains its attractiveness. 

Between the orientation of the Egyptian temples and the 
welding of space and time into space-time stretch some four or 
five thousand years of troubled history. In mathematics the two 
events appear almost contemporaneous. 

The postulational method 

Had the Greeks done nothing more than put a foundation 
under the real number system, they would have been assured of 
perpetual remembrance in mathematics. But they did a great 
deal more. Indeed, ‘Greek mathematics’ inevitably suggests 
synthetic geometry, and it was in the Greeks’ elucidation of 
spacial form that many see their greatest contribution. 

The development of geometry from a practically workable 
empiricism to a strict deductive science was extraordinarily 
rapid. The earliest proof in geometry is traditionally ascribed to 
Thales, about 600 b.c. He is said to have proved, as one of some 
half-dozen theorems, that a circle is bisected by any of its 
diameters. A century and a half later the Pythagoreans had gone 



about as far in plane geometry as students today in the first 
half of an American school course. Among other details, they 
knew the Pythagorean theorem, the properties of parallels, the 
angle-sum for any triangle and possibly for any convex rec- 
tilinear polygon, the principal facts about similar figures; and 
they had adequate geometrical equivalents for addition, sub- 
traction, multiplication, division, the extraction of square roots, 
and the Euclidean solution of x z -f- ax = a-. Some of this, 
naturally, was within the limitations implicitly imposed by their 
conception of the number system. Scarcely modified, devices of 
the Pythagorean graphical arithmetic and algebra survive in the 
techniques of our drafting rooms. 

In solid geometry, the Pythagoreans knew at least three of 
the five regular solids, and possibly all. If they did know all, 
their faith in Number as the ruler of the Cosmos may have 
suffered a setback; for the first three solids occur naturally in 
common minerals that would attract the eye of any geometer, 
while the dodecahedron and the icosahedron, having fivefold 
axes, do not occur in nature. (Copper antimony sulfide, or 
tetrahedrite, and zinc blende crystallize in tetrahedra; galena, 
rock salt, and fluorite in cubes; magnetite in octahedra. None 
are rarities.) 

But a proof that precisely five regular solids are possible 
requires a well-developed theory of Euclidean space. The proof 
is ascribed to Thcaetetus, about the middle of the fourth century 
n.c. Euclid in the same century completed the elementary theory 
of these solids in his Book XIII, as the superb climax of his 
geometry. To call these solids ‘the Platonic bodies,’ as some of 
the Greeks themselves did, not only violates history but also 
insults mathematics. It is true that Plato describes a familiar 
construction of the five regular solids from the appropriate 
regular polygons. But it is also true that he used these solids as 
pulpits from which to preach Pythagorean numerology 

With the completion of Euclid’s Elements , Greek elementary 
geometry, exclusive of the conics, attained its rigid perfection. 
It was wholly synthetic and metric. Its lasting contribution — 
and Euclid’s — to mathematics was not so much the rich store of 
*165 propositions which it offered as the epoch-making method- 
ology of it all. 

For the first time in history masses of isolated discoveries 
were unified and correlated by a single guiding principle, that 
of rigorous deduction from explicitly stated assumptions. Some 


of the Pythagoreans and Eudoxus before Euclid had executed 
important details of the grand design, but it remained for Euclid 
to see it all and see it whole. He is therefore the great perfector, 
if not the sole creator, of what is today called the postulational 
method, the central nervous system of living mathematics. 

It seems strange that Euclid’s method should have had to 
wait till the nineteenth century for the only kind of appreciation 
that counts for anything in mathematics, application. Synthetic 
metric geometry of course continued in the postulational tradi- 
tion. But this, apparently, was mere inertia; for it was decades 
after the explosive outburst of projective geometry in the nine- 
teenth century before that subject received a sound basis. And 
it was only in the a.d. 1830’s that any serious attempt was made 
to provide a postulational foundation for elementary algebra. 
Not until a.d. 1899, in the work of another great geometer, D. 
Hilbert (1862-1943, German), was the full impact of Euclid’s 
methodology felt in all mathematics. 

Concurrently with the pragmatic demonstration of the 
creative power of the postulational method in arithmetic, 
geometry, algebra, topology, the theory of point sets, and 
analysis which distinguished the first four decades of the twen- 
tieth century, the method became almost popular in theoretical 
physics in the a.d. 1930’s through the work of P. A. M. Dirac 
(a.d. 1902-, English). Earlier scientific essays in the method, 
notably by E. Mach (a.d. 1838-1916, Austrian) in mechanics 
and A. Einstein (a.d. 1878-) in relativity, had shown that the 
postulational approach is not only clarifying but creative. 
Mathematicians and scientists of the conservative persuasion 
may feel that a science constrained by an explicitly formulated 
set of assumptions has lost some of its freedom and is almost 
dead. Experience shows that the only loss is denial of the 
privilege of making avoidable mistakes in reasoning. As is 
perhaps but humanly natural, each new encroachment of the 
postulational method is vigorously resisted by some as an inva- 
sion of hallowed tradition. Objection to the method is neither 
more nor less than objection to mathematics. It may be true 
that the life-sciences, for example, are still too lush a wilderness 
for the sowing of a few intelligible postulates here and there, but 
the attempt has begun, as in the work (1937) of J. H. Woodger. 
If the Pythagorean dream of a mathematicized science is to be 
realized, all of the sciences must eventually submit to the dis- 
cipline that geometry accepted from Euclid. 



Flight from intellectual -prudery 

As Plato (430-349 b.c.) preceded Euclid (365 r— 275 ? b.c.) by 
about twenty years, it is possible but improbable that the 
geometer was influenced by the philosopher. It may be regret- 
table, but it appears to be true, that creative mathematicians 
pay little attention to philosophers whose mathematical educa- 
tion has not gone much beyond the elementary vocabulary. 

Of all changes that mathematical thought has suffered in 
the past 2,300 years, the profoundest is the twentieth-century 
conviction, apparently final, that Plato’s conception of mathe- 
matics was and is fantastic nonsense of no possible value to 
anyone, philosopher, mathematician, or mere human being. 

Not all, however, are iconoclasts of Platonic realism. Some, 
whose mathematical achievements entitle them to an opinion 
on the matter, have expressed themselves quite forcibly on the 
enduring validity of realistic mathematics. G. H. Hardy (1877-, 
English), for example, stated (1940) his belief that “mathe- 
matical reality lies outside us, that our function is to discover or 
observe it, and that the theorems which wc prove, and which we 
describe grandiloquently as our ‘creations,’ are simply the notes 
of our observations.” It will be recalled that similar beliefs 
regarding other intangibles caused some rather unpleasant 
mischief in the Middle Ages and the Renaissance. 

Plato himself may not be responsible for the more outrageous 
absurdities concerning mathematics in his dialogues; there is 
always the half-mythical figure of Pythagoras in the background. 
But it was the high poetic quality of the dialogues that preserved 
the ancient nonsense for later generations of mathematicians and 
philosophers to admire and imitate. This worked great mischief 
in geometry. In Platonic realism, the straight lines and circles 
of mundane geometry are unimportant; it is the Eternal Idea of 
a straight line or of a circle that alone is worthy of philosophic 
contemplation. Thus in this particular philosophy the useful 
abstractness of mathematics is vaporized into a nothing of 
ethereal beauty that has yet to make its first contribution to 

To a Platonic geometer it is self-evident that the Archetypal 
Circle is more rotundly round than any other curve in the 
Eternal Mind, also that no Idea is straighter than the Ideal 
Straight Line in the same everlasting locus. Hence it follows that 
terrestrial geometry should be restricted in all its constructions 


to a straightedge and a pair of compasses. If, for example, an 
angle is to be trisected, it must be done with these implements. 
It follows also that in comparison with the geometry of straight 
lines and circles, that of ellipses, parabolas, and hyperbolas is 
slightly disreputable, or at least not ideally immaculate. A geom- 
etry using any mechanical contrivances other than the sacro- 
sanct two was severely reprimanded for ‘‘thus turning its back 
on the ideal objects of pure intelligence.” 

It is not surprising that Plato disdained applied mathematics. 
In his philosophy of mathematics, Plato was the finished intel- 
lectual aristocrat, purer than the purest of pure mathematicians. 
Fortunately for both pure and applied mathematics, such an 
excess of purity, not to say prudery, did not appeal to the real 
aristocrat Archimedes. 

Archimedes by himself was an epoch in the development of 
mathematics. There is immortality enough for a dozen in his 
great discoveries, and these are overshadowed by the methods 
which he invented or perfected and which, unfortunately, 
perished with him. Centuries were to pass before science and 
mathematics overtook him. 

The legend of his life is familiar from Plutarch’s incidental 
account. He was a close friend and perhaps a kinsman of Hiero, 
tyrant of Syracuse, where he was born and where he died. During 
the siege of Syracuse by Marcellus in the second Punic w r ar, the 
mechanical armaments of Archimedes delayed and all but 
defeated the Romans. When the city fell (212 B.c.), the defense- 
less old mathematician was killed by a Roman soldier. 

'Rome'WOn the war, finally destroyed Carthage ( delenda est 
Carthago!), and marched on to almost unimaginable heights of 
splendor, but not in science or mathematics. As bluntly practical 
as the soldier who dispatched Archimedes, the Romans were 
the first wholehearted- exponents of virile living and bucolic 
thinking, and the first important people to realize that a modi- 
cum of brains can be purchased by those who have only money 
or power. When they neeaed any science or mathematics not 
already reduced to easy rule of thumb, the Romans enslaved a 
Greek. But they blundered when they killed Archimedes. He 
was only seventy-five and still in full possession of his powers. 
In the five years or more of which the soldier robbed him, his 
truly practical mind might have taught the Romans something 
to ward off the fatty degeneration of the intellect which finally 
rendered them innocuous. 

All the work of Archimedes is characterized by rigor, imagi- 



nation, and power. He may rightfully be called the second 
mathematical physicist in history, and one of the greatest. 
Pythagoras was the first. In this capacity Archimedes is almost 
unique, in that he used his physics to advance mathematics. The 
usual procedure, in which he also excelled, is the reverse. A 
sample of his great work may suffice to suggest the magnitude of 
the whole. 

In applying the method of exhaustion to the mensuration (of 
both surfaces and volumes) of the sphere, cylinder, cone, spheri- 
cal segments, spheroids, and hyperboloids and paraboloids of 
revolution, Archimedes proved himself the complete master of 
mathematical rigor and the perfect artist. Some of this involved 
(in modern notation) the evaluation of the definite integrals 

J 0 sin x dx, J’J (ax -f- x-)dx. His problem of cutting a sphere 

by a plane so that the segments shall be in a given ratio pre- 
sented him with a cubic equation of the type x z + ab- — bx-, 
which he may have solved geometrically by the intersection of 
conics. The catholicity of his interests is shown by his famous 
‘cattle problem,’ which demands incidentally the solution in 
integers x, y of x- — 4,729,494y :: = 1. Finally, in pure mathe- 
matics, Archimedes anticipated the method of the differential 
calculus in his construction of a tangent to the spiral (p = ad) 
known by his name. 

His most original work perhaps was in his applied mathe- 
matics. Here, so far as is known, he was a pioneer. Menaechmus 
and others had successfully applied the method of exhaustion 
to difficult problems (Archimedes himself mentions Eudoxus 
and attributes to Democritus the statement of the result for the 
volume of a pyramid) ; but none had applied mechanics to mathe- 
matics. Before Archimedes, no scientific mechanics existed. 
There may have been empirical rules, but such are in a different 
universe. His discover}' of the law of buoyancy practically 
created the science of hydrostatics, and his formulation of the 
theory of the lever did the same for statics. So powerful were his 
methods that he determined the positions of equilibrium and 
stability of a floating paraboloid of revolution in various posi- 
tions. True to the Greek tradition, Archimedes based his 
mechanics on postulates. His determinations of centroids were 
about as difficult as those in a course in the calculus today. For 
example, he found the centroid of a semicircle, a hemisphere, a 
segment of a sphere, and a right segment of a paraboloid of 
revolution. It is small wonder that the Moslems held Archimedes 


in almost superstitious veneration. There was not his like for two 
thousand years. 

Archimedes’ sublime disregard of convention is seen in what 
is his most curious work. It is the problem, which he solved, 
of finding the area of a parabolic segment. The proof, of course, 
is rigorous. It amounts to an integration, somewhat disguised as 
exhaustion in the official proof. It is the unofficial proof which 
is of greater interest. This came to light in 1906, when a work by 
Archimedes describing his heuristic method was found in Con- 
stantinople. To discover what the required area was, Archimedes 
translated the problem in geometry into an equivalent in mechan- 
ics. Having solved the latter, he states that the result has not 
been “ actually proved.” He then proceeds to give a geometrical 
proof in which, incidentally, he performs the first summation 


of an infinite series in history. The series is ^ 4“ ”, and he uses 


the fact that 4“" tends to zero as n tends to infinity. He had 


already summed a finite series, ^ as z . 

*= i 

An isolated gem may show that Archimedes was as per- 
spicacious as he was inventive. To the untutored mind it is 
obvious that by laying off a given segment, no matter how small, 
a finite number of times, any point on a line may be reached or 
passed. It was obvious to Archimedes only that this is an as- 
sumption which should be stated explicitly as one' of the postu- 
lates of geometry. He did so; and non-Archimedean geometries, 
in which the postulate is rejected, were constructed in the 
nineteenth and twentieth centuries. Like Euclid in his explicit 
statement of the parallel postulate, Archimedes had the true 
mathematician’s caution in the presence of the obvious. 

Modern mathematics was born with Archimedes and died 
with him for all of two thousand years. It came to life again in 
Descartes and Newton. 

Through geometry to metaphysics 

The negative obligation of mathematics to ancient philos- 
ophy has been indicated. As will appear when we discuss medi- 
eval Europe, it is possible that the obligation was reversed in 
that mathematical desert. For the present, it will be of interest to 
note how Greek mathematics was indirectly responsible for some 
profoundly interesting work in epistemology since about a.d. 




The Greek geometers left undecided four elementary prob- 
lems that were to defy mathematical ingenuity for over two 
thousand years. None of the four is of mathematical importance 
today. Historically, no more prolific problems were ever pro- 
posed, with the possible exception of Zeno’s. Repeated failures 
to settle the first three disclosed fundamental difficulties unsus- 
pected by the ancients, and necessitated a sharpening of the 
number concept. Unsuccessful attempts for about 2,300 years to 
dispose of the fourth at last suggested a great advance in mathe- 
matical methodology which now seems trivially obvious, but 
which eluded some of the keenest minds in history. 

The problems are as follows. In each of the first three, with 
due deference to Plato, the desired construction is to be per- 
formed wholly by means of a finite number of straight lines and 

Problem 1. To trisect any angle. 

Problem 2. To construct the side of a cube whose volume shall be 
twice that of a given cube. 

Problem 3. To construct a square equal in area to any given circle. 

Problem 4. To deduce Euclid’s fifth postulate from the others. 

The fifth postulate is formally equivalent to the following. Through 
any point P not on the straight line I. there can be drawn, in the plane 
determined by P and L, precisely one straight line which does not 
meet L. 

Problem 2 is equivalent to demanding a geometrical construc- 
tion, by the means prescribed, for the real root of a: 3 — 2 = 0; 
Problem 1 is similar. These two were not settled till P. L. Wantzel 
(a.d. 1814—1848, French) in 1837 obtained necessary and suffi- 
cient conditions for the solution of an algebraic equation with 
rational coefficients to be geometrically constructible in the 
manner specified. Neither of the cubics concerned satisfies the 
conditions. Thus the problems were proved to be impossible. 

If the restriction that the only permitted means are a finite 
number of straight lines and circles be removed, solutions of 
Problems 1, 2 are readily obtained, for example by conics, as 
done by the Greeks, or by linkages. The historical importance of 
these two is the impetus they gave, long after Greece, to the 
investigation of the arithmetical nature of the roots of algebraic 
equations with integer coefficients. Such roots arc called alge- 
braic numbers; a number which is not algebraic is said to be 

The third problem tapped a deeper spring. By Wantzel’s 
theorem, if Problem 3 is solvable, its algebraic equivalent must 


be a finite number of equations satisfying his conditions. The 
problem will be impossible if 7r(= 3.14 . . . ) is transcendental. 
In a.d. 1882, C. L. F. Lindemann (1852-1939, German) proved 
that 7T is transcendental. His proof, with its curious dependence 
on rational arithmetica, would have delighted Pythagoras. 

Problem 3, as 1, 2, is solvable when modified to permit the 
use of curves other than circles. The quadratrix (p, 0 polar 
equation tt p = 2 rd esc 6) invented by Hippias in the fourth 
century b.c. for the trisection problem suffices. This, however, is 
of but trivial interest; the significance of circle squaring is its 
connection with transcendental numbers. 

Squaring the circle implies an irrationality of a kind radically 
distinct from that which taught the Pythagoreans that not all 
numbers are rational; V 2 is algebraic, tt is not. It would seem a 
reasonable guess to one exploring the number system for the 
first time that all real numbers are algebraic, or at least that 
the transcendentals are extremely rare. Cantor proved in a.d. 
1872 that the algebraic numbers are the rare exceptions; the 
transcendentals are infinitely (to the power of the continuum) 
more numerous. It is an interesting' exercise to trace the implica- 
tions of the restriction to a finite number of straight lines and 
circles in the conditions of the three problems. 

Problem 4 — to prove Euclid’s parallel postulate — will re- 
appear when we follow geometry through the nineteenth century. 
It is one of Euclid’s greater achievements to have perceived that 
this postulate demands explicit statement as an assumption. 

The new quirk in methodology which finally disposed of the 
problem in a.d. 1826 may be described here, as it is one of those 
profoundly simple, powerful devices which are so obvious after 
they have once been pointed out that they are first imagined 
only by minds of the highest originality. A problem which has 
resisted the best efforts of genius for centuries may be impossible, 
or meaningless, or improperly posed. The quirk is simply to 
admit that one of these three may be the fact. This admitted, 
what seems the likeliest of 'che three is developed mathematically. 

The parallel postulate was circumvented by the third possi- 
bility: a self-consistent geometry was constructed without it. 
Problems 1, 2 evaporated when the suspected impossibility was 
pursued, and a contradiction was deduced from assumed possi- 
bility. The problem of squaring the circle suffered the same fate, 
but was much harder to dissipate. 

A brilliant application of the quirk to a modern problem was 
Abel’s proof in a.d. 1824 that the general algebraic equation of 



degree higher than the fourth is unsolvable by radicals. He 
appears to have been the first to state the methodology explicitly 
as a general procedure. As an item of historical interest, the 
Persian poet, mathematician, and connoisseur of wine, women, 
and song, Omar Khayyam, is said to have conjectured in the 
twelfth century that the algebraic solution of the general cubic 
is impossible. He was mistaken. We shall recur to this in a 
later chapter. 

Beginning about a.d. 1930, the Viennese school of mathe- 
matical logicians attacked some of the classical problems of 
philosophy, particularly metaphysics, by this methodology, 
attempting to show that the problems were either meaningless or 
improperly posed. Of course history may prove them as mistaken 
as Omar was. Needless to say, the attack was vigorously resisted, 
especially by those who refused to master enough elementary 
symbolism to enable them to read a proof in symbolic logic. 

Thus four elementary problems of Greek geometry were 
partly responsible for a subversive movement in philosophy that 
would have shocked the ancient Greek philosophers as pro- 
foundly as it shocked some of the moderns. There was at least 
a prospect in a.d. 1945 that the incipient revolution — if it may 
be called that with propriety — might necessitate some revision of 
accepted epistemology. The non-Euclidean geometry of the 
nineteenth century that issued from Problem 4 abolished Kant’s 
theory of mathematical ‘truths.’ 

Plane, solid, and linear loci 

Toward the end of Greek mathematics, a hesitant step 
toward unity and generality was taken by Pappus (probably 
third century) in his MaOrjfxanK&v <ruvccyuty&v fiifiXla. This 
collection of eight books, of which only the last six and a mere 
fragment of the second are extant, was a compendium of much 
of the mathematical knowledge of its time. The missing parts 
may have dealt with arithmetic; the six known books include 
proportion, parts of solid geometry, selected higher plane curves, 
isopcrimetric problems, spherics, centroids, special curves of 
double curvature and their orthogonal projections, and finally 
mechanics, which seems to have signified yet more geometry 
to the ingenious compiler. Those items of the collection that 
may have been due to Pappus himself have been called brilliant 
by competent critics; certainly they display a boldness of con- 
ception and an uncramped freedom of method reminiscent of 
Archimedes rather than of Euclid. If a kinematic generation of 


a curve seemed natural to Pappus, he did not hesitate to avail 
himself of it. Probably much earlier than his day it had been 
suspected that the three classic problems of Greek geometry 
are unsolvable by Euclidean methods, although no Greek mathe- 
matician is known to have stated the impossibility of Euclidean 
solutions as a working hypothesis. Accepting the suspected 
fact, Pappus proceeded to a masterly investigation of the higher 
plane curves which about five centuries of experience had shown 
to be sufficient for solving the problems. The spiral of Archi- 
medes, the conchoid of Nicomedes (second century b.c.), the 
cissoid of Diodes (same century), and the quadratrix of Hippias 
(fifth century b.c.) were accorded full geometric status. These 
outlaws of the rigid classical geometry were shown to be as 
worthy of serious attention as the hackneyed conics. The con- 
choid had been invented to solve the trisection problem, the 
quadratrix for the rectification and quadrature of the circle, 
and the cissoid for the classic Greek problem of inserting two 
geometric means between two given ‘magnitudes’ represented 
as straight-line segments. The conchoid and cissoid are alge- 
braic curves; the quadratrix is transcendental. Nevertheless, all 
three are thrown into the vaguely inclusive class of ‘linear’ loci. 
This ill-defined receptacle held all loci other than the plane and 

Circles and straight lines were ‘plane’ loci; the conics, 
‘solid’ loci, doubtless so named on account of their origin as 
sections of cones (of the second degree). In passing, one of the 
decisive achievements of Apollonius was his replacement of 
the three species of cone, used by his predecessors to obtain the 
various conics, by the right circular cone of which all are sec- 
tions. If derivation from conical surfaces was the ground for 
calling conics solid loci, it seems rather peculiar that Pappus 
should have cast the quadratrix into the nebulous limbo of 
linear loci. For two of his most striking personal contributions 
were his definitions of the quadratrix as orthogonal projections 
of certain skew curves. In one, the curve is the intersection of a 
cone of revolution and a right cylinder whose base is a spiral 
of Archimedes. Here was synthetic geometry in the grand man- 
ner almost of Archimedes himself. 

Though the classification of loci as plane, solid, and linear 
may not seem very significant to a modern geometer, neverthe- 
less it was a conscious attempt to put some system and order 
into the chaos of imaginable plane curves. Without algebraic 
symbolism, little either reasonable or useful was possible, and 



almost nothing general. If Apollonius be granted the use of co- 
ordinates claimed for him by some of his admirers, this in itself 
but emphasizes the inadequacies of the clumsy substitutes for a 
genuine symbolism used by the Greek geometers. That they 
accomplished so much that has retained its interest after twenty 
centuries or more is a tribute to their genius rather than a recom- 
mendation of their technique. But lest we overvalue our own 
acquisitions at the expense of theirs, we may remember that 
there is as yet no satisfactory classification of the uncountable 
infinity of transcendental plane curves. If such a classification 
is not a dead problem, it is a project for the analysis rather 
than the geometry of the future. 

A wrong turning? 

Greek mathematics stands as one of the half dozen or so 
supreme intellectual achievements of our race. Its best is now 
well over two thousand years behind us. Looking back on it in 
the light of the mathematics that has developed since the early 
decades of the seventeenth century, we shall try to see it dis- 
passionately in “the cold beams of the history of learning.” Its 
two greatest achievements — those traditionally attributed to 
the Pythagoreans — shine out as clearly as ever, and with them, 
Euclid’s. Apart from Archimedes, two thousand years ahead of 
his age, what of the rest ? 

For better or worse, our technical deveopment of science 
and mathematics differs radically from that of the Greeks. Their 
mathematics is intelligible to us, and any modern can appreciate 
it at what they themselves considered its true value. Ours, be- 
yond the mere rudiments, would appear to them — with the 
exception of Archimedes — as conclusive evidence of insanity. A 
straight line to them, for example, meant a finite segment capa- 
ble of prolongation; to us a straight line is defined once for all 
from minus infinity to plus infinity. 

The limited modes of Greek thought are not ours. With the 
maturing of elementary algebra in the sixteenth and seventeenth 
centuries of our era, and the introduction of analytic methods 
in tire seventeenth, mathematics in its return to number drew 
closer to Babylon, Egypt, and India than it overdid to Greece 
after the fall of Alexandria. Except for insistence on proof, our 
preferences in mathematics as in religion are more oriental than 

It may be a hard saying, but it appears to be none the less 
true, that on the long view Greek geometry was in part a tactical 


blunder. No necessity compelled Thales, Pythagoras, Euclid 
Apollonius, and all their disciples to develop the synthetic 
method exclusively. At the beginning of their arduous mathe- 
matical journey, two possible roads had been plainly indicated 
to the Greeks by their Eastern predecessors. Either by conscious 
predilection or ironic mischance they all took the same turning 
and hewed their way through tremendous obstacles to the end 
of a blind alley. The synthetic geometry of conics marks the 
end of the journey. 

Further significant progress — that which mathematics fol- 
lows today — became possible only when the harsh way was 
retraced or temporarily forgotten, and the road which the Greeks 
had passed by in the sixth century b.c. was entered in the seven- 
teenth of our era. Returning to the thought of the East, from 
which Thales and Pythagoras had started, European mathe- 
matics detoured almost completely round the territory consoli- 
dated by the Greek geometers. Resuming a march interrupted 
twenty-three centuries earlier, mathematics during and after 
the seventeenth century proceeded with incredible speed to the 
conquest of world after new world beyond the farthest reach of 
Greek thought. 

The geometry of Ptolemy’s Almagest appears to us as an all 
but superhuman effort of mathematical genius. It and the 387 
propositions in the conics of Apollonius are the masterpieces of 
the synthetic method. But the new science inaugurated by 
Galileo and Newton in the seventeenth century needed more 
than one or two masters of mathematics every four or five 
hundred years if it was to exploit its opportunities with reason- 
able speed. 

Few mathematicians who have followed the Grecian proofs in 
Newton’s Principia believe that all the propositions demon- 
strated could ever have been discovered in one lifetime by the 
methods of Greek geometry. There are limits even to the mind 
of a Newton; and we have his own word for it that he used 
analytic methods — his calculus — for discovery. The rigid syn- 
thetic proofs were devised partly to reassure himself but prin- 
cipally so that he might be understood by others. Still, it might be 
possible to claim the Principia as a monument to synthetic geom- 
etry. But not the most generous imagination would concede the 
dynamics of Lagrange, Hamilton, Jacobi, and Lie to a hypo- 
thetical application of Greek geometry, although these crea- 
tions of the eighteenth and nineteenth centuries evolved with, 
apparent inevitability from the dynamics of the Principia. And; 



last, in reference to his method which Newton saw fit to translate 
into Greek, discovery after all is more important in science than 
strict deductive proof. Without discovery there is nothing for 
deduction to attack and reduce to order. 

Returning for a moment to the sixth century b.c., we may 
try to imagine -what mathematical history might have been if 
the Greeks had taken the Babylonian highway. Like other 
‘might have beens,’ this one is futile, except possibly as it may 
indicate which of several roads we ourselves might the more 
profitably explore. 

The orientals had a more catholic taste than the Greeks for 
number. At least some of the orientals were not terrified by mere 
magnitude; Indian mythology with its millions of deities, its “tan- 
gled trinities,” and its aeons of aeons is an adumbration of the 
mathematical infinite. For that matter, theEgyptian trinity exhib- 
its some of the seeming contradictions of the modern concept of the 
infinite, with its one-one correspondence between part and v’hole; 
and the like is evident in Christian theology, the heir, not of 
dead Greek mythology, but of oriental religions. A minor but 
significant indication of the ineptitude of the Greek mind for 
mathematical analysis is the fact that for centuries it remained 
content with a system of numeration which, compared with the 
best of the oriental work, was puerile. 

It is not definitely known that the Early Greeks were ac- 
quainted with the advances and speculations in number of other 
peoples; but from internal evidence it is highly probable that 
they must have heard of them. There are too many oriental 
inclusions in Greek mathematics to make the miracle of a 
curiously partial transmission of knowledge credible. For the 
sake of our hypothesis we shall assume that the Greeks were not 
entirely ignorant of what their neighbors to the East had done. 

Had the early Greek mind been sympathetic to the algebra 
and arithmetic of the Babylonians, it would have found plenty 
to exercise its logical acumen, and might easily have produced a 
masterpiece of the deductive reasoning it worshipped logically 
sounder than Euclid’s greatly overrated Elements. The hypoth- 
eses of elementary algebra are fewer and simpler than those of 
synthetic geometry. The algebraic-analytic method in mensura- 
tion and geometry was well within the capacity of the Greek 
mathematicians, and they could have developed it with any 
degree of logical rigor they desired. Had they done so, Apollonius 
would have been Descartes, and Archimedes Newton. 

As it was, the very perfection — for its age, and for long after 


— of Greek geometry retarded progress for centuries. It was 
admired, as it merited, by the Moslems -who finally restored it in 
the Middle Ages to a forgetful Europe; and much of the genius 
that might have gone into expanding their own arithmetic, 
algebra, and trigonometry was lavished on translation and 
commentary. If there is any truth in Bergson’s elan vital, or in 
Hegel’s philosophy of history, Greek geometry was a splendid 
disaster for both. 

This, needless to say, is not the traditional conclusion. The 
superiority of purely synthetic methods over the algebraic and 
analytic, as being more intuitive, has been urged by numerous 
distinguished mathematicians, particularly of the British school, 
since the time of Newton. And we find the same contention being 
put forward in the seventeenth and nineteenth centuries for the 
superiority of the synthetic method in projective geometry over 
the analytic. No working mathematician would deny the utility 
and suggestiveness of diagrams ; but that is not the point at issue. 
It has been said that in geometry the synthetic and the analytic 
methods are like a pair of hands; and this undoubtedly is true, 
provided the geometry handled is simple enough. But inspection 
of a treatise on modern physics, or of one on partial correlation in 
statistics, even when the analysis may be described in the 
language of geometry, reveals few synthetic proofs, if any. The 
like is true of the greater part of living geometry itself. And 
Lagrange, the great master of dynamics after Newton, prided 
himself that his analytic mechanics contained not a single 

One of the most vigorous defenses of the geometrical method 
of Euclid, Apollonius, and Ptolemy is that of Thomas Young 
(a.d. 1773-1829, English), the universal genius who is remem- 
bered for his contributions to medicine, Egyptology, elasticity, 
and the wave theory of light. His arguments are frequently 
repeated even today, especially in the intermediate instruction 
of science students. By a singular historical irony, Young’s 
defense was first printed in a.d. 1800, the year which marked 
the end of the middle period of mathematics and the beginning 
of the recent. It was republished, together with a slashing attack 
on the analytic mechanics of Lagrange, in a.d. 1855, the year in 
which Gauss, the inaugurator of the recent period, died. But if 
Young after all v r as right, his was a voice crying in the wilderness 
that few appear to have heard. For better or worse, mathematics 
in the seventeenth century committed itself to analysis, and the 
Greek methods became of only historical interest. 


The European Depression 

It is customary in mathematical history to date the beginning 
of the sterile period from the onset of the Dark Ages in Christian 
Europe. But mathematical decadence had begun much earlier, 
in one of the greatest material civilizations the world has known, 
in the Roman Empire at the height of its splendor. Mathemati- 
cally, the Roman mind was crass. 

Beyond the cumbersome Roman numerals, which can be 
called a mathematical creation only by undiscriminating charity, 
the Romans created nothing even faintly resembling mathema- 
tics. They took what little they needed for war, surveying, and 
brute-force engineering from the Greeks they had crushed by 
weight of arms, and were content. When Julius Caesar reformed 
the calendar in 46 b.c., it was no Roman who proposed leap year 
with its extra day in February, but the Alexandrian Sosigenes. 
The Roman contribution to civilization was in law, government, 
and peace at the sword’s point. 

The military Pax Romana began collapsing in earnest in 
a.d. 410, when the invaders penetrated the city of the Caesars, 
and the last garrison was recalled from Britain to stand with the 
defenders against an onrushing flood of barbarians. The debacle 
of Roman grandeur came about sixty years later, and five cen- 
turies of darkness descended on Christian Europe. 

Five years after the recall of the Roman garrisons, a riot 
in the last capital of Greek learning foreshadowed the centuries 
of confusion, and marked the end of the first great epoch in 
creative mathematics. 

One of the last of the Greek mathematicians was a woman, 
Hypatia. Like her male colleagues at Alexandria, Hypatia was a 
critic and commentator rather than a creator. Her death sym- 



bolizes the end of pagan science and mathematics, and the 
beginning of an age of faith. In 415, when Hypatia died, there 
were good works more urgent than geometry and arithmetica 
to be done. The hordes from the north were in need of civilizing 
and conversion to a gentler religion. 

To the zealous tillers of this all but virgin field it seemed 
obvious that the decaying remnants of an effete Greek culture 
must first be cleared out of the way. Had not Greek intellectual- 
ism and immorality sapped the virility of Rome? Therefore 
Greek thought must be swept back into the past. As a representa- 
tive of the older enlightenment, Hypatia was a conspicuous 
obstacle in the path of the new. Encouraged by their uncom- 
promising bishop, the willing Christians of Alexandria effectively 
removed the obstacle by inducing her to enter a church, where 
they murdered her in a needlessly barbarous manner. 1 

Mathematics lived on, just breathing, in Christian Europe. 
The next significant epoch was inaugurated in the eighth century 
by the infidel followers of the prophet Mahomet. 

European mathematics from Boethius to Aquinas 

Before passing on to the one thing of any suggestiveness for 
the development of mathematical thought that may have had a 
root in the sterile centuries, we must propitiate tradition by 
doing honor to the learned Europeans of that period whose 
names adorn the classical histories of mathematics. From a long 
list of historical celebrities we select the following as a fair sam- 
ple, with their names, their dates, all a.d. henceforth, and the 
places where they flourished: Boethius (c. 475-524, Rome, 
Italy); Isodorus (c. 570-636, Seville); the Venerable Bede (c. 
673-735, England); Alcuin (735-804, born at York, labored in 
France); Gerbert (950-1003, Rome); Psellus (1020-1100, Greece, 
Constantinople); Adelard (early eleventh century, England); 
Robert of Chester (early twelfth century, England, Spain). This 
list may be very considerably lengthened without adding any 
undue burden of mathematics. 

No census of the leading European mathematicians of the 
Middle Ages would be complete without the memorable name of 
Thomas Aquinas (1226-1274, Naples, Paris, Rome, Pisa, 
Bologna). Although this Newton of scholastic theology is not 
usually counted among the elite of medieval mathematics, we 
shall see that he might be. 

In contemplating the barren record from Bede to Aquinas, it 
is well to remember that while European civilization rotted, 



another culture, the Moslem, 2 was conserving the Greek classics 
and developing the algebra and arithmetic of India in prepara- 
tion for the European Renaissance. We are immediately con- 
cerned only with the contributions of Christian scholars. 

Of the European background, it is sufficient to recall the 
persevering struggle of the church to dominate the people and 
mildly educate a few of them, and the dawning enlightenment 
that accompanied the crusades of the eleventh to the thirteenth 
centuries. The crusades no doubt accelerated throughout a 
wakening Europe the diffusion of knowledge that began with the 
Moslem conquest of Spain in 711. These influences are reflected 
in a gradual change, beginning in the twelfth century, in the 
character of European mathematics. 

Medievalists disagree on which century, the twelfth or the 
thirteenth, was of greater significance in the awakening of 
Europe. The distinction, if any, is unimportant for mathematics. 
There was one item of any moment, and only one. Latin versions 
of Greek mathematical classics, made for the most part from 
translations by the Moslems into Arabic or Persian, became 
available to European scholars. While remembering with grati- 
tude the devoted labors of the translators, we need not forget 
that translation is not creation. The best of the translations 
added nothing new to mathematics; the worst, by men who 
might be erudite scholars but who were wretched mathemati- 
cians, added only misunderstanding. 3 

To see how low mathematics sank, and to guess how low it 
may sink again if the enthusiasts for all things medieval prevail, 
we resume our sample from Boethius to Aquinas, and note what 
some of these giants did. In their own semicivilized times, several 
of the men cited were conspicuous conservers of such civilization 
as there was. Some helped to found elementary schools, others 
taught, while the more thoughtful wrote shabby textbooks and 
zealously cultivated theological numerology. Before the great 
depression got well under way, Boethius described the consola- 
tions of philosophy in a homily which was to solace many in dire 
need of solace in the Middle Ages. Gerbcrt, one of the more 
enlightened popes — unjustly accused at one time of collaboration 
with the Devil 1 — donned the tiara in 999 and steered the church 
safely through that ominous year 1000 whose widely heralded 
Satanic disasters unaccountably failed to materialize. In defer- 
ence to scholarship, it must be recorded in passing that one school 
of medievalists proves conclusively that no disasters were ever 
prophesied, while an equally positive school proves conclusively 


that they were. Whatever the facts, Gerbert wrote on division 
and on computation by the abacus, collected trifles on polygonal 
numbers, compiled an alleged geometry from Boethius and 
another still less enlightened source, and is said to have had a 
part in popularizing the Hindu numerals. He is also reputed to 
have been a man of vast learning and acute intellect. Some of his 
letters reveal him as singularly dense in the most elementary 
arithmetic. If Gerbert’s contributions to mathematics are passed 
over in silence, it is for the sufficient reason that he made none, 
despite the fact that no history of mathematics is complete 
without his illustrious name. The like applies to Bede and 
Alcuin , 5 justly reckoned among the pedagogical heroes of the 
Middle Ages. Climaxing this phase, Psellus seems to be remem- 
bered chiefly because it is doubtful whether he ever did any- 
thing at all in mathematics. His introductions to Nicomachus 
and Euclid, perhaps fortunately for his reputation as a mathe- 
matician, are of uncertain authenticity. But his version of the 
quadrivium persisted through the fifteenth century. 

With Adelard and Robert of Chester, we advance to the next 
stage. An indefatigable traveler and painstaking scholar, Adelard 
was an intelligent collector and translator of mathematical 
classics. The path of the bibliophile in the eleventh century was 
less smooth than it is now, and Adelard frequently risked his 
skin to secure his coveted manuscripts. He is credited with one 
of the first European translations of Euclid into Latin and with a 
translation of Al-Khowarizmi’s astronomical tables. Adelard’s 
one putative original contribution to mathematics was an utterly 
trivial problem in elementary geometry. Robert translated 
Al-Khowarizmi’s algebra. 

All of Adelard’s predecessors together managed to keep some 
semblance of life in the rudimentary mathematics of Christian 
Europe. Beyond that, the best of these worthy men made only 
clumsy calculations in the simplest arithmetic, or attempted to 
approach elementary geometry in a spirit that would have dis- 
graced a Greek schoolboy of fourteen. The mathematical awak- 
ening of Europe was due to no effort of theirs, and their illustrious 
names might be dropped from the history of mathematics with- 
out loss. But tradition, rightly or wrongly, forbids. We therefore 
continue our descent to the nadir of mathematics, and follow the 
learned Boethius into the abyss. 

It was the elementary schoolbooks of Boethius that set the 
mathematical pace of the Middle Ages in Europe. Returning to 



the Pythagorean synthesis, Boethius expounded a denatured 
quadrivium of arithmetic, music, geometry, and astronomy. 
Considered apart from their medieval content, the names of 
these four divisions of the Pythagorean tetrad are impressive. 
But could Pythagoras have looked behind the names, he might 
have been somewhat disappointed. The geometry, for example, 
made a brave show by starting from Euclid. But it did not get 
very far. Only the enunciations of the propositions in Book I 
and a few in III, IV were offered to the eager students. At the 
lowest ebb of mathematical intelligence, the liberally educated 
graduated from geometry when they had learned by rote the 
enunciations of the first five propositions in Book I of the Ele- 
ments. Later, when more of Euclid became available, ambitious 
would-be clerics were encouraged to memorize the proofs of these 
propositions. The fifth, appropriately enough, was nicknamed 
the Asses’ Bridge ( pons asinoruin ) . Few attempted the hazardous 
passage over the equal angles at the base of an isosceles triangle. 

In his arithmetic, Boethius followed the Alexandrian 
Nicomachus. As we have seen, Nicomachus in his turn had 
followed Pythagoras at his most mystical, producing a shoddy 
treatise on the elementary properties of numbers that might have 
been composed by an amiable philosopher with a passion for 
numerology. In passing, mathematicians rather deprecate such 
effusions being called “the theory of numbers” in some of the 
traditional accounts . 7 To confuse astrology with astronomy 
would be less wide of the mark. However, Boethius reproduced 
the sieve of Eratosthenes and offered some amusing trifles on 
figurate numbers. Proof seems to have had no greater attraction 
for him than for his master Nicomachus. Boethius also is 
credited by some controversialists with a problematical intro- 
duction of the Hindu numerals to supplement the abacus and 
the counting board of trade. The practical outcome of all this 
was a cumbersome reckoning sufficient for simple transactions 
involving money, and for keeping the calendar in order so that 
the date of Easter might not elude annual recapture. To call any 
of this computation — or of the debased geometry — mathematics 
is a gross exaggeration. The significance of mathematics as a 
deductive system had been forgotten. Science having sunk to the 
level of superstition, the other half of the Pythagorean vision 
survived only in the fantastic absurdities of sacred and profane 
numerology. Number indeed ruled the darkened universe of the 
European Middle Ages. 


The erudite Boethius however made another contribution to 
learning that may have had more influence on the development 
of mathematics than all the editions of his sorry arithmetic and 
geometry ever had. He made a part of Aristotle’s system of logic 
available to European scholars in a Latin translation. 

What follows is only a speculation. But even so, it is less 
depressing and possibly less futile than the dreary record of the 
homunculi mathematici whose saintly lives and lack of works 
constitute the official history of mathematics in Christian Europe 
of the Middle Ages. 

Submathematical analysis 

To the scientific or mathematical mind up to the twentieth 
century, the logical disputes which absorbed a major part of 
the mental energy of the Middle Ages had long seemed the acme 
of futility. But in the two decades following the world war of 
1914-18, all things medieval became more popular than at any 
time since the rise of modern science. For sufficient reasons, 
many had lost their illusion of progress. Blaming science, with 
an occasional diatribe against mathematics, disillusioned ideal- 
ists groped blindly back to the twelfth century, or even the 
ninth, seeking an authoritative assurance of a security more 
satisfying than any science. Should followers of these hopeless 
travelers to the past equip themselves properly, they may restore 
to the future a treasure-trove for the history of mathematical 
thought comparable to that which has been recovered from 
Babylon. They may discover where European mathematics 
went underground after the death of Hypatia, and what shape it 
assumed during its long burial. 

Two things in particular may be sought with some prospect 
of reward: the struggle of the Greek philosophical concept of the 
infinite, attributed to Anaximander in the sixth century b.c., to 
get itself transmuted into the modern mathematical infinite; the 
closely cognate struggle of mathematical analysis to get itself 

The mathematical mind was not dead in the Middle Ages. It 
was merely sleeping. In its uneasy rest it imagined something 
curiously like mathematics; but it was powerless to throw off its 
dreams and wake. The theological subtleties and the scholastic 
quibblings which absorbed the intellect of generation after 
generation of potential geometers and analysts were the troubled 
dreams of a torpid mathematics. 



Possibly the suddenness with which the mathematical mind 
awoke after its long sleep will seem a less abrupt discontinuity 
when scholars shall have had the patience to explore medieval 
thought in the light of modern mathematics. Somewhere and 
somehow in the Middle Ages mathematics suffered a profound 
mutation. When it went to sleep, mathematics was Greek; when 
it awoke, it rapidly developed into something that was not 
Greek. The Moslems were not responsible for the change. Their 
mathematics, with its shunning of the infinite, is no closer in 
spirit to analysis than is that of the Babylonians. 

The transition from ancient thought to modern appears to 
have been more difficult in mathematics than it was in science. 
There seems to be no clear-cut instance of a mathematical mind 
in the late Middle Ages two or three centuries ahead of its time, 
as Roger Bacon’s (1214—1294?) was in science. Even Bacon, all 
but fully awake in science, was still as fast asleep in the mathe- 
matics which he eulogized as were any of his European opponents 
of the thirteenth century. His mathematical reasoning 8 is still 
that of the Aristotelian scholastics whom he believed he was 
confounding with his logic: “I say therefore that if matter can 
be the same in two substances, it can be the same in an infinite 
number . . . Therefore matter is of infinite power. Wherefore 
also of infinite essence, as will be proved, and therefore it must be 
God” — which, of course, Bacon is refuting. He seeks to accom- 
plish his purpose by an argument based on a Euclidean postulate 
which the sharper scholastics had rejected. They did not assume, 
as did he, that “the whole exceeds any of its parts” is valid for 
infinite assemblages. 

The similarity between Greek art, from sculpture to architec- 
ture, and Greek mathematics has often been remarked. There is 
no need to pursue this resemblance here; either it is felt as more 
than a vague metaphor, or it is dismissed as having no possible 
meaning. A like comparison between Gothic architecture and 
modern mathematics had impressed many not primarily inter- 
ested in mathematics before O. Spengler exploited his Faustian 
theory of mathematics since the Greeks. Thus, writing in 1905, 
Henry Adams said of the cathedral at Chartres that “Chartres 
expressed ... an emotion, the deepest man ever felt — the 
struggle of his own littleness to grasp the infinite.” 9 

Adams also gives a sympathetic parody of the medieval per- 
version of elementary mathematical reasoning to the uses of 
scholasticism, in an imaginary but convincing debate between 


those two formidable champions of submathematical analysis 
Abelard (1079-1142) and William of Champeaux (1070-1122)! 
But as a specimen of Gothic mathematics at its best, his quota- 
tion from Archbishop Hildebert (eleventh century) is more 
suggestive: “God is over all things, under all things, outside all; 
inside all; within but not enclosed; without but not extended; 
above but not raised up; below but not depressed; wholly 
above, presiding; wholly beneath, sustaining; wholly without, 
embracing; wholly within, filling.” This goes far beyond any logic 
of Aristotle; it all but makes its subject the Excluded Middle. 

Lest, after this, the project of disinterring mathematics from 
the dialectic of the Middle Ages seem fantastic, two most signifi- 
cant facts may be recalled to encourage those who would proceed. 
Georg Cantor (1845-1918, Germany), founder of the modem 
theory of the mathematical infinite, was a close student of 
medieval theology. In this connection, it appears that the tradi- 
tionally religious type of mind is the most strongly attracted by 
Cantorian mathematics. The other significant fact is the dis- 
covery 10 by K. Michalski (Polish) in 1936 that William of Occam 
(1270-1349, English) proposed a three-valued logic, thus antici- 
pating to a slight extent the work of non-Aristotelian mathe- 
matical logicians in many-valued logics since 1920. Aristotle’s is 
a two-valued logic, the Values’ being ‘truth,’ ‘falsity’ assigned 
to propositions. In Occam’s logic Aristotle’s excluded middle is 

To the gentle and scholarly Boethius with his translation of 
Aristotle belongs a large share of whatever credit there may be 
for having put mathematics to sleep in medieval Europe. The 
rest may be awarded to the hordes of tireless logicians who 
strove for centuries to weld theology and philosophy into a self- 
consistent whole. When at last Thomas Aquinas (1227-1274) — 
“the dumb ox of Sicily,” as he was called by jealous and envious 
rivals, but master of them all — succeeded, interest in the stupen- 
dous project had already waned. Roused by the waspish Mos- 
lems, Christian Europe woke, and turned, possibly with a sigh of 
relief, to some science and mathematics. 

Glancing ahead, we recall that in the first week of September, 
1939, the medieval mind at last came into its own once more in 
Christian Europe. It would be interesting to know what our 
regenerated descendants will remember of our science and 
mathematics in 2039, and what people, if any, are to be the 
Moslems of the future. 


Detour through India, Arabia, 

and Spain 

A.D. 400-1300 

The sudden rise and the almost equally sudden decline of the 
Moslem culture in the seventh to the twelfth centuries is one of 
the most dramatic episodes in history. 1 Here we are interested 
only in seeing what enduring influence the culture of this period 
had on mathematics; and we must not let the sudden brilliance 
of Mahometan civilization, contrasted against darkened Europe, 
dazzle us into seeing more in Moslem 2 mathematics than was 
actually there. 

By 622 the followers of Mahomet were well started on their 
travels. Their swarming under the green banner was the greatest 
religious revival on record, its only close competitor being the 
counter-revival of the crusades in the twelfth and thirteenth 
centuries with their avowed purpose of supplanting the banner 
by the cross. From the capture of Damascus in 635, the victori- 
ous Moslems proceeded to the siege of Jerusalem, taking that 
holy city in 637. Four years later they had subdued Egypt, 
incidentally putting the final touches to the destruction of the 
Alexandrian library. This, however, was only a youthful indis- 
cretion, as the Moslems were shortly to settle down and become 
the most assiduous patrons of Greek learning in history. Having 
subdued Egypt, they next (642) took Persia and all its civilized 

Seventy years later (711), the conquerors entered Spain, 
where they furthered civilization for about eight centuries before 



being expelled by the Europeans they had at last stung awake. 
In addition to sowing the fertile seeds for centuries of war, they 
had brought the arithmetic and algebra of India and Greece, 
and Greek geometry, to Europe. Bagdad on the Tigris under the 
Abbasid caliphs from 750 to 1258 became the capital of culture 
in the East, Cordova in Spain the intellectual queen of the 
West. After the Moors’ defeat (1212), Jewish scholars — many of 
whom had acquired their learning from the tolerant Moslems — 
vied with Christian teachers in spreading the science and mathe- 
matics that were to relegate scholasticism to the limbo of 
forgettable but unforgotten misadventures in intellection. 

The final act was delayed till 1936, when the degenerated 
followers of the Prophet returned triumphantly to Spain under 
a red and gold banner, to harry the descendants of the people who 
had driven out their ancestors some three and a half centuries 
before. During the Moslems’ long absence, Spain had con- 
tributed nothing to mathematics. With the involuntary depar- 
ture of the Jews in the late fifteenth century, savage intolerance 
for all free thought, whether of Jew or gentile, succeeded 
sane liberality, leaving four sterile centuries as its monument 
to science. 

Partial emergence of algebra 

Perhaps the most significant advance of the period was the 
gradual emergence of algebra as a mathematical discipline in its 
own right, all but independent of arithmetic and geometry but 
closely affiliated with both. 

Trigonometry also became clearly recognizable as a separate 
division of mathematics; and some see in the trigonometry of the 
Moslems their greatest and most original work. Part of its 
originality may be granted. But for reasons that will appear as 
we proceed, the trigonometry is not comparable in importance 
for living mathematics with the Hindu-Moslem algebra, with 
its frustrated struggle toward operational symbolism. Before 
trigonometry could function vitally in modern mathematics 
it, like geometry, had to become analytic. There is no hint of 
such a transformation before the seventeenth century, and 
actually it was fully accomplished only in the eighteenth. The 
Moslem trigonometry is still essentially Ptolemy’s, amplified and 
refined by some algebraic reasoning and an extensive application 
of Hindu-Moslem arithmetic to the computation of tables. From 
its very nature as mathematics of the discrete, algebra could not 


become a province of mathematical analysis; and hence it was 
beyond disturbance by the analytic upheavals of the seventeenth 

Moslem algebra appears to have evolved from the late Greek, 
as in Diophantus, and the much sharper technique of the Hindus. 
Estimates of Indian algebra differ widely, but on two points there 
is substantial agreement. Proof was as distasteful to the Indian 
temperament as it was congenial to the Greek; the Hindus were 
as apt in calculation as the Greeks were inept. Only by an ag- 
gressively sympathetic scrutiny of Hindu algebra can anything 
resembling proof be detected. Rules were clearly stated, but the 
statement of rules is not proof. A third feature of early Hindu 
algebra strikes a modern observer as extremely curious: the first 
skillful algebraists seemed to find indeterminate (diophantine) 
equations much easier than the determinate equations of elemen- 
tary algebra. The reverse is the situation today. 

A small sample from Hindu algebra will suffice here to indi- 
cate the quality of what the Moslems inherited, conserved, and 
partly spoiled. In the sixth century, Aryabhatta summed arith- 
metical progressions, solved determinate quadratics in one 
unknown and indeterminate linear equations in two unknowns, 
and used continued fractions. Shortly after, Hindu algebra 
experienced what some consider its golden age, with the work 
of Brahmagupta in the early seventh century, just as the Mos- 
lems were about to start on their travels. Brahmagupta stated 
the usual algebraic rules for negatives, obtained one root of 
quadratics, and, most remarkable, gave the complete integer 
solution of ax ± by — c, where a, b, c are constant integers. He 
discussed also the indeterminate equation ax- + 1 = y-. The 
last is misnamed the Pellian equation; it inspired Lagrange in 
1766-9 to some of his greatest work in pure mathematics. It is 
fundamental in the arithmetical theories of binary quadratic 
forms and quadratic fields. Its place in the history of mathema- 
tics will be noted presently. 

Again it seems strange that algebraists who did not hesitate 
to attack problems of real difficulty failed to see completely 
through simple quadratics. As remarked in connection with 
Eudoxus, the early algebraists were halted by a deficiency in the 
Greek logical faculty. Without an extended number system, it 
was impossible for the Hindus to create much that even resem- 
bled a scientific algebra. Thus Mahavira in the ninth century 
unhesitatingly discarded as inexistent the imaginaries he en- 


countered, without attempting to account for their appearance. 
Three centuries later, Bhaskara recognized that formalism 
produces two roots for quadratics, but rejected the negatives. 

Hindu algebra, however, took a hesitant step toward opera- 
tional symbolism. To what has already been noted regarding 
symbolism, the following summary of the principal advances 
of the Hindus toward symbolic algebra may be added. Critics 
disagree on how far the Hindus got in this direction, but what 
follows seems to be established fact. Aryabhatta (sixth cen- 
tury) suggested the use of letters to represent unknowns. 
Brahmagupta (seventh century) used abbreviations for each 
of several unknowns occurring in special problems, also for 
squares and square roots. A negative number was distinguished 
by a dot; and fractions were written in our way, but without the 

bar, thus, 

3 . 


A manuscript assigned to the period 700-1100 

displays a cross, like our plus sign, written after the number 
affected, to indicate minus. Bhaskara (twelfth century) imitated 
Brahmagupta in the notation for fractions, also in the custom 
of putting one member of an equation under the other, and in a 
systematic, syncopated script for successive powers. There was 
no sign for equality. Brahmagupta also effected the reduction of 
Diophantus’ three types of quadratic equations in one unknown 
to the standard form now current. 

Differences of opinion concern the weight to be given these 
devices. By the most liberal estimate, the Hindus had the gist 
of algebraic symbolism as an operational technique proceeding 
according to fixed rules and in standardized patterns: the tech- 
nique for solving problems of certain types was indicated in the 
mere writing of the problems. All the elaborate verbal directions 
for taking the successive steps toward a solution are explained 
away as insurance against stupidity. For even at its best, Hindu 
algebra, in spite of its free use of abbreviations, was still largely 
rhetorical in that operational directions were not fully symbol- 
ized. The least generous appraisal admits no advance in method- 
ology beyond Diophantus. The Hindus themselves appear to 
have left no record supporting the first estimate. Possibly they 
imagined the meaning of what they did so obvious as to render 
comment on the methodology superfluous. Introspection in 
mathematics is a modern neurosis. 

There is also the vexed question of how much of the Hindus 
algebra was their own and how much Greek. Until competent 


scholars reach some shadow of agreement, there is little point in 
others reproducing their divergent conjectures. With the dis- 
covery of Babylonian algebra, the dispute seems less likely than 
ever to be settled in a finite time. There are also — a fascinating 
possibility — the ancient Chinese to be considered. Did they, or 
did they not, influence the Sumerians ? Possibly the Sumerians 
taught the Chinese? Perhaps the Indians taught them all: Or 
did they all teach the Indians? And what part did Syria play? 
One type of argument supporting a favored conjecture may be 
noted in passing. If civilization A is assumed to be older than 
civilization B , and if in B a certain type of problem was discussed 
at a later date than in A , it follows that B got the problem from 
A. This is according to the diffusion theory of culture. ■* On the 
alternative spontaneous theory, no conclusion can be drawn; and 
even on the diffusion theory it remains to be shown that the data 
are uncontaminated by intrusions of spontaneity. The scarcity of 
documentary evidence complicates the problem. 

Fortunately for our immediate concern, these profound ques- 
tions need not be settled before the influence of Hindu arithmetic 
and algebra on Moslem mathematics can be substantiated. The 
Moslems themselves admit having translated Hindu works. It 
is therefore reasonable to infer that the Moslems were influ- 
enced by the Hindus. Assuming this, we note once more the 
human propensity to take the longest way home. Like the Greeks 
in their indifference to Babylonian algebra, the Moslems finally 
turned their backs on the rudimentary hints of an operational 
symbolism in the Hindu algebra and in their own, and wrote out 
everything, even the names of numbers, in full. The Moslem 
retrogression in this respect was as long a backward step as any 
in the history of mathematics. Absorbed in the intelligent collec- 
tion and painstaking examination of numerous interesting speci- 
mens, they missed the main thing completely. Only in 1489, in 
Germany, with J. W. Widmann’s invention of -r, — , did algebra 
begin to become more operationally symbolic than it had been 
for Diophantus and the Hindus. 

Before leaving Hindu algebra, we note what is usually con- 
sidered its high tide. We shall inspect this rather closely, first 
in the haze of a golden sunset, then in the unsentimentalized 
light of mathematics. The two appearances are strangely dis- 
similar; it may be left to individual taste which is preferred. 

Bhaskara, about 1150, “gave a method of deducing new sets 
of solutions of Cat 5 -4-1 = y- from one set found by trial.” 5 The 


problem is that of the so-called Pellian equation: to solve 
Cx 2 + 1 = y 2 in integers x, y, where C is a given nonsquare 
integer and xy ^ 0. Bhaskara also discussed Cx 2 + B = y 2 , C 
B being non-square integers. His very elementary devices have 
excited the liveliest admiration. Contemplating them in the 
golden haze, we observe 6 that “the first incisive work [on the 
Pellian equation] is due to Brahmin scholarship,” and note that 
this equation “has exercised the highest faculties of some of our 
greatest modern analysts.” We also see that Bhaskara’s attack 
on the equation “is above all praise; it is certainly the finest 
thing which was achieved in the theory of numbers before 
Lagrange.” 7 

The first of these three quotations is a verifiable statement 
of fact. The second seems to imply that Bhaskara’s “incisive 
work” is qualitatively comparable to that of “some of our 
greatest modern analysts.” The third makes Bhaskara’s tenta- 
tive, partial solution a finer thing in the theory of numbers than 
Euclid’s direct, complete solution of x 2 + y 2 = z 2 . 

In the unflattering light of mathematics, it appears that 
Bhaskara could find any number of solutions provided he was 
lucky enough to guess one. He possessed no means of determining 
whether a given Pellian equation was solvable. Nor, even when 
he had derived further solutions from a lucky guess, could he tell 
whether he had all solutions. His process for generating solutions 
from an initial one was ingenious. But he ignored the only points 
of any difficulty or mathematical interest: the existence of a solu- 
tion and the completeness of those solutions exhibited. 

In contrast with this supreme achievement of Brahmin 
scholarship, we observe what happened when one of “our great- 
est modern analysts” exercised his “highest faculties” on the 
Pellian equation. Lagrange admitted that he had to stretch him- 
self to accomplish what he did. In 1766-9 he settled the problem 
of existence and gave a direct, nontentative method for find- 
ing all solutions. Bhaskara was an empiricist; Lagrange, a 

The unromantic conclusion is that Bhaskara fell far below 
the standard set by Euclid, a standard which was not reached 
again till Lagrange attained it in the eighteenth century. It 
seems not unjust to draw the same conclusion regarding the rest 
of Hindu mathematics. But it is generally conceded that the 
better Hindu algebraists were far ahead of Diophantus in 
manipulative skill. This, and their frustrated attempt to create 


an operational symbolism, appear to be the chief contribution 
of the Hindus to the development of mathematics. 

Moslem algebra seems to have hesitated between the tastes 
of Greece and India, choosing the latter in its most creative 
period, only to lapse into an impossible rhetoric as it became 
classic in the ninth century with the masterpiece of its most 
famous exponent, Al-Khowarizmi. 

Indian algebra was translated into Arabic and Persian by 
the Moslems; and as Arabic was an important language not only 
in scholarship but also in commerce and war, Greek and Indian 
algebra, simplified and somewhat systematized by the Moslems, 
at last penetrated Europe. If the crusades of the twelfth and 
thirteenth centuries did nothing more, they helped indirectly to 
spread algebra, trigonometry, the classics of antiquity, and 
contagious diseases. 

Of an impressive list of Moslem translators, commentators, 
and minor contributors, only two need be mentioned here. Each 
showed some originality and both, particularly the first, pro- 
foundly influenced early European algebra. To give him almost 
his full name, Mohammed ibn Musa Al-Khowarizmi (died c. 
8S0) of Bagdad and Damascus produced the first treatise (c. 
825) in which occurs an equivalent for our ‘algebra’ — al-jebr 
ttfalmuquabala , meaning ‘restoration and reduction.’ The refer- 
ence is to what now would be called transposition of negatives to 
yield equations with all terms positive, and to subsequent reduc- 
tion by collecting like powers of the unknown. This appears to 
have been Al-Khowarizmi’s own idea; the work as a whole is a 
compost of Greek and Hindu results. His principal advance in 
the positive direction was an application of the Hindu number- 
names to the numerical solution of equations. 

Al-Khowarizmi’s signal progress in the negative direction 
has been noted. 8 Why he returned to a purely rhetorical algebra 
unenlivened by any trace of symbolism, seems not to be known. 
A psychiatrist might say it was the death instinct having its 
way. All but strangled, algebra survived, thereby demonstrating 
that more than a resolute attempt at suicide is necessary to 
deprive mathematics of its life. In what is reputed to be an 
algebraic masterpiece, Al-Karkhi (c. 1010) continued in the 
rhetorical tradition. If the fact were not well established, it 
would be difficult to believe that medieval European algebraists 
had the persistence to find out what the rhetorical algebraists 
of Islam were attempting to communicate. 


Whether justly or not, algebra without symbolism is rather 
disappointing to the average layman who has been assured that 
“the Arabs invented algebra.” Unfortunately, an expert knowl- 
edge of Arabic was never one of the more graceful accomplish- 
ments of a gentleman, nor even of a scholar or a country squire, 
as was a slight acquaintance with Latin or Greek in the eight- 
eenth century. Consequently the mathematician or the his- 
torian of mathematics competent to form a personal estimate of 
Moslem algebra has always been a rarity; and of the few who 
have deigned to share their findings with those innocent of 
Arabic, some have presented the outcome of their researches 
in the familiar symbolism of algebra as taught to beginners 
today. In certain respects those sophisticated versions of the 
original verbiage resemble beggars masquerading in robes of 
satin. To appreciate the difference between the original and 
its modern disguise, the curious should prevail upon a profes- 
sional scholar of Arabic to read them a verbatim translation of 
an original document in Moslem algebra. In lieu of this, the 
excerpt presently transcribed from the English translation of 
Al-Khowarizmi’s Algebra by F. Rosen (1831) may be exhibited. 
Rosen’s translation reproduces the original Arabic, so that 
cognoscenti may savor its quality. The passage quoted is from 
A history of mathematical notations (1928) by the American 
historian of mathematics, Florian Cajori. Rosen remarks that 
“numerals are in the text of the work always expressed by 
words: Hindu-Arabic figures [numerals] are only used in some 
of the diagrams, and in a few marginal notes.” The excerpt 

What must be the amount of a square, which, when twenty-one dirhems 
are added to it, becomes equal to the equivalent of ten roots of that square? 
Solution: Halve the number of the roots; the moiety is five. Multiply this by 
itself; the product is twenty-five. Subtract from this the twenty-one which arc 
connected with the square; the remainder is four. Extract its root; it is two. 
Subtract this from the moiety of the roots, which is five; the remainder is three. 
This is the root of the square which you required and the square is nine. Or 
you may add the root to the moiety of the roots; the sum is seven; this is the 
root of the square which you sought for, and the square itself is forty-nine. 

Of course symbolism of itself is not mathematics, and no 
amount of beautifully appropriate notation can make shoddy 
or trivial reasoning look like mathematics. Extensive tracts 
of mathematics contain almost no symbolism, while equally 
extensive tracts of symbolism contain almost no mathematics. 


However, as in the above specimen, the total avoidance of 
symbolism is not always a virtue to be imitated, especially by 
neophytes. For laymen who may have difficulty in recognizing 
algebra when it is spread before them, Rosen transposes Al- 
Khowarizmi’s rhetorical exercise into its symbolic equivalent: 

* 2 + 21 = lOx; 

X = W + Vim 2 - 213 = 5 + V(2 5 - 21), 

= 5 + V4 = 5 + 2 - 3, 7. 

This particular equation recurs many' times in the early history 
of algebra. 

Al-Khowarizmi’s treatise is credited with a large share of the 
mathematical awakening of Christian Europe; and a twelfth- 
century Latin translation of a lost tract of Al-Khowarizmi’s on 
the Hindu numerals is said to have done much to acquaint Euro- 
peans with that great invention. Giving this labor of transmis- 
sion its full historical weight, and balancing it against Moslem 
algebra, we may leave the reader to find his own point of equilib- 
rium somewhere among the three following estimates of Moslem 

“The greatest mathematician of the time [early ninth cen- 
tury], and, if one takes all circumstances into account, one of 
the greatest of all times was Al-Khowarizmi.” In the next two, 
full weight is given to Moslem trigonometry, to be described 
presently. “Their [the Moslems’] work was chiefly that of trans- 
mission, although they developed considerable originality in 
algebra and showed some genius in their work on trigonometry.” 
“If the work produced [by the Moslems] be compared with that 
of Greek or modern European writers it is, as a whole, second- 
rate both in quantity and quality.” 

Three centuries after the great Al-Khowarizmi had finished 
his labors, and therefore toward the close of the cultural period, 
the Persian poet-mathematician, Omar Khayyam (died c. 1123), 
reached a considerably higher mathematical level than any of his 
predecessors. This devil-may-care, somewhat cynical philosopher 
had imagination. Not content with collections of rules, Omar 
classified cubic equations and devised a method of geometrical 
solution for numerical cubics, general within the limitations of 
the existing number system. Others, said to have taken the hint 
from Archimedes, had solved cubics by the use of conics long 
before Omar; indeed, the method was familiar to the Moslems 
of the ninth century. It was not Omar’s technical labors, how- 


ever, but his erroneous conjectures that cubics are algebraically, 
and quartics geometrically, unsolvable, that mark him out as 
more than a faithful transmitter and a skilled tactician and 
algebraic taxonomist. 

But bold and original as he was, Omar steadfastly refused to 
accept negative roots. His hyperbolas, too, were deficient in 
negative branches. Once more it was the failure to come to grips 
with the number concept that thwarted both algebra and 

The emergence of trigonometry 

In its literal meaning of ‘triangle measurement,’ trigonometry 
is as old as Egypt, of course in an extremely rudimentaryform. 
Greek astronomy demanded spherical geometry, and this, com- 
bined with the reduction of observations, necessitated what we 
should call the computation of trigonometric functions. 

Ptolemy in the second century after Christ summarized in his 
(jxeydcXr] crbvrafi'i =) Almagest the main features of spherical trig- 
onometry, and indicated a method for the approximate calcula- 
tion of what amounts to a crude table of sines, or ‘half-chords.’ 
Ptolemy used chords; the crudity was unavoidable by the geo- 
metrical method necessitating interpolations over too wide an 
interval. Thus, traditionally, plane trigonometry was merely a 
computational adjunct to spherical trigonometry, and hence the 
mathematically more important elements of trigonometry 
emerged with unnecessary slowness. Perhaps the ultimate source 
of the Hindu and Moslem development of trigonometry was not 
applications to surveying but the astronomical necessity for 
sharper interpolation. 

A Hindu work of about the fourth century advanced con- 
siderably beyond Greek trigonometry in both method and 
accuracy, giving a table of sines calculated for every 3.75° of 
arc up to 90°. The rule used to compute the table was erroneous, 
but possibly it gave results of sufficient accuracy for the inexact 
observations of the age. In any event, its reversion to empiricism 
affords an interesting illustration of the radical distinction 
between the Greek mathematical attack and the oriental or, for 
that matter, between the oriental and the modern even at its 
most crudely practical. 

The Moslems adopted and developed the Indian trigonom- 
etry. Their first notable advance was due to the astronomer 


Al-Battani (died 929) in the ninth century. If not actually the 
first to apply algebra instead of geometry exclusively to trigo- 
nometry, this astronomer-mathematician was the earliest to 
take a long stride in that direction. In addition to the Hindu 
sine, he used also the tangent and cotangent. Tables for the 
last two were computed in the tenth century, when also the 
secant and cosecant made their appearance as named trigo- 
nometric ratios. As the concept of a function was still about six 
hundred years in the future, none of his work bears much 
resemblance to elementary trigonometry as it is today. 

Three more names may be cited as marking definite stages in 
the emergence of trigonometry as a distinct mathematical dis- 
cipline. Abul-Wefa in the latter half of the tenth century began 
the systematization of all the trigonometry known at the time, 
and reduced it to a decidedly loose deductive system. The first 
Moslem text on trigonometry as an independent science was 
that of the Persian astronomer Nasir-Eddin (1201-1274). The 
book was more than a mere compendium, giving abundant 
evidence of a sure mathematical talent. Like the algebra of Dio- 
phantus, this work fell too close to the end of its cultural epoch 
to exert its full weight on the future of mathematics, and Euro- 
peans duplicated much of it without, apparently, being aware of 
its existence. 

The last name we shall cite suggests a curious bit of history 
which might be worth exploring. Leonardo of Pisa (Fibonacci) 
will reappear in the sequel; here we note that he published his 
masterpiece, the Liber abaci , in 1202 (revised, 1228). Leonardo 
was largely responsible for acquainting an awakening Europe 
with the Hindu-Moslem algebra and the Hindu numerals. 
Among other significant trifles in Leonardo’s book is the well- 
known algebraic identity 

(a 2 + b")(c- + d 2 ) = (ac ±bd ) 2 - f (ad + be) 2 . 

It would be interesting to know where Leonardo picked this up 
on his travels in the East, as it is easily shown to include the 
addition theorems for the sine and cosine. It also became a 
germ of the Gaussian theory of arithmetical quadratic forms, and 
later of interesting developments in modern algebra. With the 
appropriate restrictions as to uniformity, continuity, and initial 
values when a,b,c,d are functions of one variable, the identity 
contains the whole of trigonometry. 


Mathematics at the crossroads 

While Europe slept and all but forgot Greek mathematics, 
the Moslem scholars were industriously translating all they could 
recover of the works of the classic Greek mathematicians. 
Several of these translations became the first sources from which 
Christian Europe revived the mathematics it had all but let die. 
For this timely service to civilization, the Moslems no doubt 
deserve all the gratitude they have received. But even at the 
risk of appearing ungracious, any mathematician must temper 
gratitude with the hard fact that scholarship and creation are in 
different universes. 

Had the Moslems done nothing but preserve and transmit, 
they would scarcely have merited a passing mention in even the 
briefest account of the development of mathematics. This may 
seem too brutually direct; yet, by the only standard whose main- 
tenance insures progress rather than stagnation, it is just. The 
one criterion by which mathematicians are judged is that of 
creation. Unless a man adds something new to mathematics he 
is not a mathematician. By this standard, the Moslems were not 
mathematicians in their extremely useful work of translation 
and commentary. Remembering that we are interested chiefly 
in things that have lasted, we shall consider briefly the Moslems’ 
work of translation and commentary, and with it some of their 
trigonometry, in the light of living mathematics. 

Only a few specialized historians of mathematics ever really 
digest any of the Greek masterpieces. Not only is life too short 
for those who would acquire some usable mathematics for them 
to master Apollonius or Archimedes; it would also be as wasteful 
an effort as could be imagined. Nothing could possibly come out 
of it but erudition. The works of the Greek masters were washed 
up centuries ago on the banks of the living stream; the spirit of 
their essential thought, and some few results that beginners learn 
today more easily by modern methods, alone survive. Thus the 
Mosem contribution of translation and commentary has lasted, 
not mathematically, but only as a monument to scholarship. It 
can be argued that without this moribund mass of Greek mathe- 
matics, there would have been no inspiration for the new mathe- 
matics of the seventeenth century: without Apollonius there 
would have been no Descartes, without Diophantus no Fermat, 
and so on, and on. Against this it can be maintained that origi- 
nality was smothered under a blanket of erudition, and that, 


barring sentimental reservations, the greatest service the dead 
past can render mathematics is to bury its dead. The arguments 
are equally incapable of objective decision; so we shall leave 
them with the fact that mathematicians no longer study the 
Greek classics preserved by the Moslems, nor have they for the 
past two and a half centuries. 

The fate of spherical trigonometry, on which the Moslems 
lavished so much of their skill, illustrates the inevitable recession 
of things first developed for immediately practical ends. Their 
utility may remain; but any living scientific interest they may 
once have had has long since died. Spherical trigonometry is no 
longer in the living stream of mathematics, either for its content 
or for any method it may have. Unless a student today requires 
the subject for some definite routine, such as the old-fashioned 
positional astronomy, he need not even know that spherical 
trigonometry exists. Of all the subjects in elementary mathe- 
matics, spherical trigonometry is probably the deadest and the 
most repulsive to anyone with the faintest stirrings of a feeling 
for vital mathematics. At rare intervals some optimistic enthu- 
siast attempts to breathe a little life into the dry bones; but 
after a few perfunctory rattles, silence descends once more, and 
spherical trigonometry is deader than ever. Even the profound 
revision (1893) of the entire subject in terms of nineteenth- 
century algebra and analysis by E. Study (1862-1922, German) 
attracted only passing attention from mathematicians. 

Plane trigonometry with the Greeks and the Moslems was 
encouraged principally because it was a useful servant to its 
elderly spherical sister, and she in her turn was honored mainly 
for her sendees to astronomy. While plane trigonometry was 
growing up, astronomy was cbiefest of the sciences and the only 
one demanding any considerable application of mathematics. 
Astronomy then needed only the solution of triangles. When 
positional astronomy receded to a subordinate routine in modern 
astronomy, the trigonometric functions — for reasons in no way 
connected with the solution of triangles — became the indispensa- 
ble mathematical aid from celestial mechanics to spectroscopy. 

As modern science evolved after Galileo, astronomy became 
but one science of many, some being of perhaps even greater 
practical importance than astronomy in a scientific civilization. 
Here again it was the trigonometric (or circular) functions that 
were to prove indispensable, and again for no reason even 
remotely concerned with the solution of triangles. The sine and 


cosine derive their scientific importance from two properties: 
they are the simplest periodic functions; they furnish the first 
instances of a set of orthogonal functions. Both properties were 
centuries in the future when the Moslems had finished their 
work; the second had to wait for the integral calculus. Orthog- 
onality underlies modern applications of the sine and cosine, 
making possible, as it does, the solution of important boundary- 
value problems arising from the differential equations of mathe- 
matical physics. 

It would be difficult to imagine a physical science without 
orthogonal functions; but our successors may, and they may then 
look back on us as we look back on our predecessors. May they 
be as mindful of us as we are of the Hindus and the Moslems for 
what they invented, developed, and passed on to us to be further 

As we take leave of the Moslems, we see them hesitating 
rather forlornly at the crossroads of ancient and modern mathe- 
matics. Progress passed them before they could make up their 
minds to turn their backs on the past they had rescued from 


Four Centuries of Transition 


The thirteenth to the sixteenth centuries in Europe is one of 
the most eventful periods in world history'. These four centuries 
also include the sharply marked transition from ancient mathe- 
matics to modern, die break being clearly discernible in the 
half-century following 1550. As will appear in the Italian solu- 
tions of the cubic and quartic equations of about 1545, algebra 
then was still in the Greek-Hindu-Moslem tradition. The French 
work (Vieta) of the latter half of the sixteenth century' was in a 
totally different spirit, and one which mathematicians today' can 
recognize as akin to their own. In less than fifty' years, the Greek 
and middle oriental traditions became extinct in creative 
mathematics. 1 

The precise dates 1202, 1603 in the heading are intended 
merely to recall two of the most significant landmarks in the 
four centuries of transition. The first marks the publication of 
Leonardo’s Liber abaci; the second, the death of Vieta, the first 
mathematician of his age to think occasionally as mathemati- 
cians habitually think today'. 

The somewhat narrow scope of Vieta’s technical achieve- 
ments is irrelevant to his importance in the development of 
mathematics. It was not what he actually accomplished in 
mathematics, although that was considerable, that counted; it 
was the quality' of his thought. Whether or not the mathe- 
maticians of the early seventeenth century' consciously looked 
back on Vieta as their herald, he was. They quickly surpassed 
what he had done; but their superiority' was one of degree, not 
of kind. 



Mathematics was ripe for the transition a full two centuries 
before ft actually happened. The sharp change was delayed by a 
social chaos in which civilization had all it could do to keep 
alive. Concurrently, deeper movements were sweeping the super- 
ficial barbarism of the times back into the past, and clearing the 
way for a more humane economy in which mathematics shared. 
A brief recapitulation of the main events responsible for the 
delay and the subsequent advance will make the transition seem 
less miraculous than it otherwise might. Wc shall point out in 
passing what some of the major events implied for the future of 

Opposing currents 

All the learning of the ancient world could not continue to 
flood into Europe for much longer after 1200 without setting up 
opposing currents. Broadly, the conflict became a struggle be- 
tween established authority to maintain its vested interests un- 
impaired, and a quickening impatience with mere authority as 
the final arbiter between free inquiry and dictated belief, whether 
in knowledge of the natural universe or in government and reli- 
gion. With both the rapid assimilation of the ancient learning 
and "the process of the suns” the thoughts of men were being 
broadened, and nothing short of complete destruction of the race 
could halt progress. As it happened, disaster was averted by a 
rather narrow margin. 

On the side of liberality, the universities of Paris (1200), 
Oxford, Cambridge, Padua, and Naples were founded between 
1200 and 1225. Although the early universities bore but little 
physical remcmblancc to what they later became, they were 
extremely significant steps toward intellectual freedom. The 
thirteenth century also saw the founding of the great orders of 
the Franciscans and the Dominicans, at least part of whose 
activities were educational. 

A too strict devotion to scholasticism in the early universi- 
ties precluded any serious study of mathematics; but the phe- 
nomenon of thousands of eager students at Paris squatting in 
mildewed straw and avidly absorbing Abelard’s (1079-1142) 
hair-splitting dialectics and his humanistic contempt for mathe- 
matics shows at least that the capacity for abstract thought was 
not extinct. Some of the universities being direct outgrowths oi 
the cathedral schools, it was but natural that they should favor 
the curricula they did. As late as the fifteenth century, only a 



smattering of arithmetic and a few propositions of Euclid satis- 
fied the mathematical demands of a liberal education as certified 
by a bachelor’s degree from Oxford. 

Throughout all this period, war was a capital industry. The 
sack of Constantinople by the Crusaders in 1204, barbarous 
though it was in itself, might be reckoned with the cultural gains, 
as a convincing demonstration that greed and religion form a 
highly explosive compound. There appears to be no doubt, how- 
ever, concerning what finally issued from the Holy Inquisition, 
established in one of its milder forms in 1232, shortly after the 
Moors were disciplined in Spain. Nor is there great difference of 
opinion in democratic countries about the protracted dissolution 
of feudalism, and the faint hints of democracy in the conse- 
quent rise of the middle and merchant classes in the two cen- 
turies following 1250. Although civilization almost dissolved in 
the process, the decay of feudalism and the gradual concretion of 
national monarchies accelerated the growth of knowledge after 
the critical period was safely passed. 

The confusion and intolerance already evident in the thir- 
teenth century became worse confounded in the fourteenth. But 
the picture was not painted wholly in one color. The names of 
Dante (1265-1321) and Petrarch (1304—1374) suggest that an 
occasional ray of light penetrated the gloom; while that of 
Boccaccio (1313-1375) recalls that some could still appreciate a 
bawdy story even in the presence of the Black Death (1347- 
1349), which carried off between a third and a half of the popula- 
tion of Europe. The science which less than three centuries in the 
future was to be fought with all the weapons of intolerance has 
alone wiped out such plagues. 

In this fateful century the Hundred Years’ War also got well 
under way, lasting from 133S to 1453 by one count, and from 
1328 to 1491 by another. Whichever estimate is correct, there 
seems to be general agreement that war flourished somewhat 
rankly in Christian Europe for nearly two centuries. For ruthless 
brutality, cynical disregard of the pledged word, and unblushing 
degeneracy, the famous Hundred Years of the fourteenth and 
fifteenth centuries had no superior till the twentieth. Not the 
will, but only the lack of adequate means of destruction, pre- 
vented a complete return to barbarism. It seems incredible that 
anything faintly resembling civilization could survive such a 
reversion to brutchood. But it did. Had some poet of the blackest 
years sung “the world’s great age begins anew,” as Shelley did 


shortly before the deepest squalor of the Industrial Revolution, 
they would have called him mad. 

One great scientific invention in the first half of this same 
fourteenth century of our era passed almost unnoticed except as 
a curiosity for a few careless years, when Europeans quite 
suddenly envisioned the limitless horizons of destruction revealed 
by gunpowder. Radical improvements in the art of war conse- 
quent on this warmly appreciated gift of alchemy were to 
necessitate much refined pure mathematics and higher dynamics 
in the accurate calculation of trajectories. Without the mathe- 
matics of exterior ballistics, old-fashioned gunpowder, or even 
modern high explosive shells and rockets, would be less effective 
than the bows and arrows of the English archers at Agincourt. 
It therefore seems unlikely that war could be abolished by 
suppressing mathematics. 

Although it could not have been foreseen at the time, the 
fifteenth century was to prove a landmark in mathematics as it 
was in all knowledge. In 1453 Constantinople fell to the Turks, 
and Eastern culture found its most hospitable welcome in Italy. 
The powerful family of the Medici in this period rendered dis- 
tinguished service to civilization by their patronage of scholars 
and collectors of manuscripts. So far as mathematics is con- 
cerned, the net gain of this liberality was a further increase in 
erudition. But something of infinitely greater importance than 
the accumulation of libraries happened at this time, and at last 
made mathematics accessible to anybody with the capacities to 
take it. About 1450 the printing of books from movable types 
started in Europe. 

In the first fifty years of European printing, Italy alone 
produced about 200 books on mathematics. During the next 
century the output was slightly over 1,500. The majority of 
course were elementaiy textbooks; but when a work with some 
real mathematics in it was printed, it became public property 
instead of the choice possession of a few who could afford a 
handmade copy. This was the second of the three major advances 
in the dissemination of mathematics. The first has already been 
noted in the transition from oriental secrecy to Greek free 
thought. The third was delayed for nearly four hundred years 
after the second, until 1826, when the first of scores of low-priced, 
high-grade periodicals devoted exclusively to mathematical re- 
search appeared. Printing also furthered mathematics through 
its economic insistence on a uniform, simplified symbolism. 



Toward the close of this century, the discovery of America 
(1492) implied possibilities for mathematics that nobody could 
have predicted. The necessity for accurate navigation in mid- 
ocean, and the determination of position at sea by tables based 
on dynamical astronomy, indicate the connection between 1492 
and Laplace’s celestial mechanics completed only in the first 
third of the nineteenth century. Some of the fundamental work 
(Euler’s) of the eighteenth century in the lunar theory was 
undertaken to meet the need of the British Admiralty for re- 
liable tables. The stimulus for these particular advances, origi- 
nating in the voyages of Columbus and others, was about evenly 
divided among exploration, land grabbing, commerce, and the 
brutal struggle for naval supremacy. From Laplace’s develop- 
ment of the Newtonian theory of gravitation in his dynamical 
astronomy, issued the modern theory of the potential and much 
of the analysis of the partial differential equations of physics 
in the nineteenth and twentieth centuries. Thus a modern mathe- 
matician, whether he lives in the United States or in China, who 
devotes his life to problems in potential theory with increasingly 
bizarre boundary conditions owes part of his livelihood indirectly 
to Columbus. So also does a mathematical physicist who com- 
putes perturbations in atomic physics; for the theory of pertur- 
bations was first elaborated in dynamical astronomy. 

The sixteenth century was equally pregnant with great 
things for the future of mathematics. The names of Leonardo 
da Vinci (1452-1519), Michelangelo (1475-1564), and Raphael 
(1483-1520), three of the foremost among a host, will recall 
what this critical age, the century of Copernicus (1473-1543), 
was in art; while those of Torquemada (1420-1498), Luther 
(1483-1546), Loyola (1491-1556), and Calvin (1509-1564) may 
suggest what it was in the higher things of life. Cardan (1501 — 
1576) published (1545) his Ars magna, the sum and crown of 
all algebra up to his time, only two years after Copernicus, on 
his deathbed, received the printer’s proofs of his epoch-breaking 
Dc revoluttonibus orbium codestium. 

The impact of Copernicus’ work on all thought and on all 
social institutions is too familiar to require comment here. 
Mathematically, the Copernican theory was not a complete re- 
jection of Ptolemy. The circular orbits of the Greeks re- 
mained, also thirty-four of Ptolemy’s seventy-nine epicycles, 
and the sun itself had a small orbit. Although Aristarchus had 
anticipated the heliocentric theory of the solar system, Coper- 


nicus was profoundly original in his provision of a reasoned basis 
for what had been only a prophetic conjecture. If any one man 
is to be remembered as the precursor of modern mathematical- 
physical science, Copernicus has as good a claim as any. 

Two further advances stand out as portents of the mathe- 
matics and science that were so shortly to become recognizably 
like our own. Stevinus (Simon Stevin, 1548-1620, of Bruges) is 
usually considered by physicists as the outstanding figure in 
mechanics between Archimedes and Galileo. Among other 
things, he stated (1586) the parallelogram of forces in the equiva- 
lent form of the triangle, and gave a complete theory of statical 
equilibrium. Modern statics is usually said to have originated 
with Stevinus. He also had as clear ideas of fluid pressure as was 
possible without the integral calculus. Incidentally, it may be 
noted that the development — such as it was — of mechanics 
by the predecessors of Stevinus is less readily evaluated than is 
the concurrent progress in mathematics. Reconsideration of 
medieval contributions to statics has usually deflated the first 
excessive claims in behalf of some more or less obscure writer 
to have anticipated Stevinus and even, on occasion, Galileo. 
Such was the case, for instance, with the suddenly inflated 
reputation of Jordanus Nemorarius (first half of thirteenth 
century) as a mechanist of high rank. It is now the opinion of 
mathematicians and physicists who have had the patience to 
sift his rhetoric that Jordanus was as unintelligent as his con- 
temporaries in his conception of mechanics. 

The other man of science in this period whose work was to 
influence mathematics indirectly but profoundly two centuries 
after his death was William Gilbert (1540-1603), physician to 
Queen Elizabeth of England. Except for some of its attempts 
at theory, Gilbert’s De magnete (1600) was a thoroughly scien- 
tific treatise on the behavior of lodestones and other magnets. 
After the consequences of Newtonian gravitation had been 
elaborated, A. M. Ampere (1775-1836, French), Gauss, G. 
Green (1793-1841, English), and others in the first half of the 
nineteenth century created the mathematical theory of magnet- 
ism. Either the subject was inherently more difficult than gravi- 
tation, or less able mathematicians attacked it. Twice as long 
was required to breach it as had been needed for gravitation. 
Of course the immediate utility of the Newtonian theory may 
have enticed the leading mathematicians of the eighteenth cen- 
tury away from Gilbert’s work; and there is the human possi- 


lx 3 

bility that the dynamics of inaccessible heavenly bodies appeared 
as a grander project than the attraction of magnets that could 
be weighed in the hand. Thus Laplace gave the sublimity of 
celestial mechanics as his chief reason for devoting his life to 
it, and he was by nature anything but sentimental. However, he 
did not always mean all he said. 

All of these men of science of the sixteenth century are over- 
shadowed in mathematical significance by one not usually 
reckoned nowadays with the professional mathematicians. 
Thirty-six years of Galileo’s (156T-1642, Italian) life fell in this 
period of transition from ancient mathematics to modern. As 
the universally recognized founder of modern science, Galileo 
influenced all mathematics, pure and applied. 

There was thus no lack of scientific daring in the sixteenth 
century', whatever the traditional custodians of consciences may 
have thought of it all. It may be said here once for all that some 
of the custodians did not always look kindly' on young science 
struggling to free itself from the fetters of authoritative tradition, 
scholastic as well as ecclesiastic. Human nature being what it is, 
there is nothing remarkable in any' of this. Both sides believed 
they' were right; and the side which had all of the power save 
that of indomitable courage believed that it also had all the 
right. Conflict was inevitable. It -was savage enough, but no 
more so than its belated echo in the third and fourth decades of 
the twentieth century', when science in some of the European 
states once more found itself fighting for its existence. 

As for the hostility' to science in the period of transition from 
ancient to modern thought, it is a mistake to blame one Christian 
sect rather than another. Anyone who cares to search the 
record may verify for himself that shades of creed were not the 
fundamental difference between those ivho welcomed science 
and those who sought to drive it out. The dissension lay' deeper, 
in the ageless and irreconcilable antagonism between old minds 
and y'oung, between those who can accept change and those w'ho 
cannot. In the sixteenth and seventeenth centuries, the y'ounger 
mind finally won its freedom and retained it for over two hun- 
dred years. During those brief centuries of free thought, science 
and mathematics prospered, and life for the majority on that 
account was less indecent than it was in the days of the Black 
Death and the Hundred Years’ War. 

It would be astonishing if mathematics had failed to respond 
to the crosscurrents of so tempestuous a transition from the old 


to the new as the four centuries from 1200 to 1600. But for the 
greater disasters for which human stupidity was only partly 
culpable, the response might have come two centuries earlier 
than it did. 

A terminus in algebra 

All through this period, as in the preceding, geometry con- 
tinued to stagnate. Beyond translations of the Greek classics, 
such as the Latin editions of Euclid (1482) and Apollonius 
(1537), the work in geometry did not rise above the level of what 
would now be exercises in elementary textbooks. The Greek 
methods appeared to be exhausted, and mathematical progress 
was wholly in the divisions of arithmetic, algebra, and trig- 
onometry. At the beginning of the period, arithmetic and algebra 
were still confused in a loosely coordinated alliance; at the end, 
they were satisfactorily divorced. Trigonometry also gained its 
liberty from astronomy in this period. 

The Liber abaci (1202) of Leonardo of Pisa (c. 1175-c. 1250) 
has already been mentioned. This famous book by a man who 
was not by training a scholar at last converted Europe to the 
Hindu arithmetic. Leonardo himself is better known in mathe- 
matics by his other name, Fibonacci (son of Bonaccio). The son 
of a warehouse official, Fibonacci traveled for business and 
amusement in Europe and the Near East, observing and analyz- 
ing the arithmetical systems used in commerce. 

The obvious superiority of the Hindu numerals and the 
Hindu-Moslem methods of computation inspired Fibonacci’s 
book; and in spite of outraged protests from conservative mer- 
chants and the then-equivalents of chambers of commerce, the 
abacus and the counting board were finally (about 1280) rele- 
gated to the attic in European trade. Thus Fibonacci is indirectly 
responsible for the deluge of practical manuals on elementary 
computation and the flood of commercial arithmetics which 
have poured from the printing presses of the world ever since 
the fifteenth century. In spite of their great practical utility, 
none of these indispensable works has contributed anything of 
importance to the development of mathematics. 2 

Fibonacci also expounded the Eastern algebra with genuine 
understanding, but otherwise made no advance. His Practica 
geometriae (1220) gave an equally enlightened treatment of 
elementary geometry. His original work lay in the borderland 
between arithmetic and algebra. In his Liber quadratorum 



(c. 1225), Fibonacci discussed some special diophantine systems 
of the second degree, such as x- + 5 = y 2 , x- — 5 = z-, which 
arc harder than they look. Judged by the standard set by Euclid 
in his integer solution of x 2 + y 2 = z~, Fibonacci’s work is on a 
far lower level. It does not seem to have occurred to him that 
the real problem in diophantine analysis is to find all solutions, 
not merely some. This failure to sense the generality of a problem 
is characteristic of the distinction between ancient and modern 
algebra, also between mathematics and empiricism. Euclid was 
the one exception in about two thousand dreary years of a de- 
based theory of numbers; even Diophantus, as we have seen, 
was content with special cases. 

Although it is distinctly a minor issue, we must mention 
Fibonacci’s famous recurring series defined by 

“b ^n, w 0, 1, . . . , Ho ~ 0, — 1, 

which gives the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... . Fibonacci 
encountered this sequence in a problem (which the reader may 
recover for himself) concerning the progeny of rabbits. There 
is an extensive literature, some of it bordering on the eccentric, 
concerning these numbers and their simplest generalization, 
tt„ + » = au n+ i + bu n , a , b constant integers, the most interesting 
modern work being that inaugurated by E. A. Lucas (1842-1891, 
French) in 1878. Some professorial and dilettant esthetes have 
applied Fibonacci’s numbers to the mathematical dissection of 
masterpieces in painting and sculpture with results not always 
agreeable, although sometimes ludicrous, to creative artists. 
Others have discovered these protean numbers in religion, 
phyllotaxis, and the convolutions of sea shells. 

It would be interesting to know who first imagined anything 
transcendental in Fibonacci’s numbers. Their simplest origin 
is in the Greek problem of dividing a line in extreme and mean 
ratio, the so-called golden section. It is said that some of the 
measurements of Greek vases, also the proportions of temples, 
exemplify the golden section; and one prominent psychologist 
even claimed to have proved that the pleasure experienced on 
viewing a masterpiece alleged to be constructed according to the 
golden section is a necessary consequence of the solid geometry 
of the rods and cones in the eve. 

Fibonacci’s quality as a mathematician emerges unmistak- 
ably in two isolated items, which also hint at the delay in the 
development of mathematics consequent on the social chaos 


that followed him. The first is his use of a single letter, in one 
instance, to denote a number in his algebra. Possibly this is the 
earliest definite trace of the generality of algebra as distinguished 
from mere syncopation and verbally expressed rules for numeri- 
cal computation. 

The second item marks Fibonacci as a true mathematician 
far ahead of his time. Being unable to give the algebraic solu- 
tion of x 3 + 2x 2 + 10a; = 20, Fibonacci attempted to prove 
that a geometrical construction of a root by straightedge and 
compass alone is impossible. He could not have succeeded with 
what was known at his time. He then proceeded to find a numeri- 
cal approximation to a root. There was nothing in algebra like 
the inspiration for the attempted proof of impossibility till the 
nineteenth century. 

All through the period of transition, algebra was concerned 
principally with the solution of equations. After quadratics had 
been solved within the limitations of the existing number system, 
the central problem was to find similar, that is, ‘radical,’ so- 
lutions for the cubic and quartic equations in one unknown. 

There are two distinct problems in the solution of algebraic 
equations: to construct, by means of only a finite number of 
rational operations and root-extractions, performed on the 
literal coefficients of a given equation, all functions of those 
coefficients which shall reduce the equation to an identity; to 
construct a numerical approximation to a root of an equation 
with numerical coefficients. The first problem is called the solu- 
tion by radicals, and is the one of greater interest in the develop- 
ment of algebra. The problem of approximating to a root is the 
one of importance in applications, for two reasons. As will appear 
considerably later, solution by radicals is impossible for the 
general equations whose degree exceeds four; the explicit radical 
solutions of the cubic and quartic are all but useless in numerical 

The problem of approximation is said to have been effectively 
solved by the Chinese in the thirteenth and fourteenth centuries 
a.d. This work, if authentic, excelled most of what Europeans 
accomplished in numerical solutions 3 until almost the same pro- 
cedure was reinvented by W. G. Horner (1773-1827, English) in 
1819. Unfortunately, like nearly all oriental mathematics except 
Indian arithmetic and algebra, the Chinese method might as well 
never have been invented for any influence it had on the develop- 
ment of mathematics. Neither in the Orient nor in Europe did it 



start a forward movement, and it cannot be said to have passed 
into the living stream. Its human interest is the evidence it 
affords (if authentic) that mathematical talent is not the exclu- 
sive possession of any one race or of any one people. 

The solution of the cubic and quartic by radicals was essen- 
tially completed by 1545. (The implied reservation refers to the 
lack of understanding at the time of negative and imaginary 
roots.) The history of this final triumph is spiced by a violence 
and chicanery that seem a trifle excessive even for the unin- 
hibited sixteenth century. Cardan (1501-1576), whose name 
ornaments the solution of the cubic in every intermediate text- 
book on algebra, obtained the solution from Tartaglia under 
promise of secrecy and published it as his own in the Ats viagna 
(1545). This ingenious algebraist is also renowned as an astrol- 
oger, his masterpiece in that direction being a horoscope of 
Christ. Among others of his excesses which offend our squeamish 
modern taste, Cardan is said to have disciplined a wayward son 
by cutting off his ears. 

Tartaglia (Nicolo, 1500-1557, Italian; the nickname Tar- 
taglia means ‘the stutterer’) had intended — so it is said — to 
crown a projected work of his own with his solution of the cubic. 
His name commemorates a split palate, which the traditional 
account credits to a saber slash inflicted by an inefficient soldier 
who, in doing his duty, was assisting in the massacre of the 
inhabitants of Tartaglia’s native Brescia. They had taken 
refuge in the local cathedral. Tartaglia, a boy of twelve at 
the time, was left for dead but, owing to the devotion of his 
mother and of the dogs who licked his wounds, recovered. As 
a mature man, Tartaglia contributed to the obsolescence of 
sabers by pioneering work in exterior ballistics, investigating 
(1537) the range of a projectile and stating that the range is 
greatest when the angle of projection is 45°. 

The solution of the quartic also harmonized vrith its social 
background. Its hero, Ferrari (1522-1565), was less fortunate 
than Tartaglia in his family affections. He is said to have been 
poisoned by his only sister. But Italy in the Renaissance with- 
out arsenic would be like veal without salt. 

In recounting these traditional embellishments of Italian 
algebra in the sixteenth century, we have tried merely to show 
that mathematics can live and flourish in what purists might 
call barbarism, and to suggest that it may survive beyond 2000. 
1 here has been no intention of misjudging the protagonists in 


the drama of algebra by a misapplication of our own domestic 
ethics. As we live according to our lights, they live according 
to theirs; and if their domestic relations now seem strangely 
foreign to us, we have but to glance at modern international 
relations to feel thoroughly at home. On the domestic front, 
theft of scientific work is still practiced, unfortunately for 
struggling young men. 4 

The scientific history of the cubic and quartic is involved 
and possibly not yet thoroughly unraveled. It appears to be as 
follows. Scipio del Ferro (1465-1526) of the University of 
Bologna solved x 3 + ax = b in 1515, and communicated the 
solution to his pupil, Antonio Fior, about 1535. Tartaglia then 
solved x 3 + px- = q and rediscovered Fior’s solution. Not 
believing that Tartaglia could solve cubics, Fior challenged 
him to a public contest, in which Tartaglia solved all of Fior’s 
equations while Fior failed to solve any of Tartaglia’s. 

Cardan was not a lame copyist. He was the first to exhibit 
three (real) roots for any cubic. He advanced beyond the mere 
formal solution in recognizing the irreducible case (all roots real), 
when the radicals appearing are cube roots of (general) complex 
numbers. The first to recognize the reality of the roots in the 
irreducible case was R. Bombclli in 1572. Cardan also suspected 
that a cubic has three roots, although he was baffled by nega- 
tives and imaginarics. His most important advance, however, 
was the removal of the term of the second degree. This had an 
element of scientific generality which, apparently, he failed to 
appreciate fully. 

Ferrari’s solution of the quartic (c. 1540) also appeared in 
the Ars magna. The solution is substantially the same as that 
in textbooks on algebra, leading to a cubic resolvent. 

These solutions of the cubic and quartic mark a definite end 5 
in the algebraic tradition of Diophantus and the Hindus. They 
were sheer tours de force of ingenuity. Modern mathematics 
deprecates mere ingenuity and seeks underlying general princi- 
ples. Proceeding from a minimum of assumptions, a modern 
mathematician exhibits the solutions of particular problems as 
instances of a general theory unified with respect to some concept 
or universally applicable method. An isolated solution obtained 
by ingenious artifices is more likely to be evidence of incomplete 
understanding than a testimonial to perspicacity. The ingenious 
solver of special problems may still be a useful member of 
mathematical society, in that he turns up mysterious phenomena 



for abler men to strip of their mystery, but he is no longer 
regarded as a mathematician; and to call a man who imagines 
he is a mathematician a problem solver is to offer him the unfor- 
givable insult. This is one line of cleavage between ancient and 
modern mathematics. 

It may be asked why these solutions of the cubic and quartic 
arc still included in the second school course in algebra. Nobody 
with any sense attempts to use them for numerical computation, 
and as encouragements to skillful trickery they are positively 
detrimental. Instead of these relics of the sixteenth century it 
would be more to the purpose to offer the unified treatment of 
quadratics, cubics, and quarries which has been available since 
1770-71, when Lagrange set himself the problem of seeing why 
the ingenuities of Tartaglia, Ferrari, and their successors worked. 
Such a treatment is considerably easier than that certified by 
historical tradition. 

A beginning in algebra and trigonometry 

Passing by an enormous mass of minor contributions, such as 
short tables of binomial coefficients, anticipations of Pascal’s 
arithmetical triangle, improvements in algebraic notation, and 
the like, we come to the first notable progress toward generality 
in the methods of algebra and trigonometry, and hence in all 
mathematics. The slow accretion of details which ultimately 
passed into elementary mathematics in permanent if modified 
form is not comparable in importance for the whole of mathe- 
matics with the striving for uniformity in methods. Success in 
this direction created a science from what had been little better 
than a museum of tricks that worked occasionally, although 
nobody understood or seemed to care why. 

The transition from the special to the general is first un- 
mistakably discernible in the work of Vieta (Francois Vietc, 
1540-1603, French), who, like Fibonacci, was not a mathe- 
matician by training or profession. Vieta’s activities ranged 
from cryptography in the military service to politics. At various 
times he was a member of parliament in Bretagne, a master of 
requests at Paris, and a king’s privy councillor. His mathematics 
was his recreation. Some have stigmatized his work as prolix and 
obscured by a private jargon. But even if this were true, such 
superficial defects could not conceal the essential qualities of 
generality and uniformity in much of what Vieta invented. 

There is no need to describe Victa’s contributions in detail. 


His attack on quadratic, cubic, and quartic equations brings out 
the essentials. In each case he removed the second term of the 
equation by a linear transformation on the unknown. Cardan had 
already done this for the cubic; Vieta appreciated the importance 
of this step as a general procedure. For the cubic he made a 
second rational transformation, and finally a simple one involv- 
ing a cubic irrationality. This produced a quadratic resolvent; 
and hence the solution of the cubic was reduced to the essential 
steps which the nineteenth-century theory of equations was to 
prove unavoidable by any device. 

In this work we see also a germ of the theory of linear trans- 
formations, whose ramifications were to branch out all through 
later algebra and thence, in the concept of invariance, through 
mathematics as a whole. Here also we see a clear recognition of 
the art of reducing an unsolved problem to the successive solu- 
tion of problems already solved, and the beginnings of tactical 
uniformity and generality. Vieta’ s solution of the quartic was 
similarly scientific, and led to the familiar cubic resolvent. 

In Vieta’s work we observe again the curious retardation of 
the number system. Negative roots appear to have been unin- 
telligible to him, although he noted the simplest of the relations 
between the coefficients of a given equation and the symmetric 
functions of its roots. He also considered the possibility of resolv- 
ing the polynomial /(*) in an algebraic equation f(x ) = 0 into 
linear factors. Anything approaching completeness or proof in 
this direction was far beyond the algebra of the time, and in 
fact was not attained till Gauss in 1799 settled the matter by 
giving a proof which would be admitted 6 today for the funda- 
mental theorem of algebra. 

Letters had been used before Vieta to denote numbers, but he 
introduced the practice (c. 1590) for both given and unknown 
numbers as a general procedure. He thus fully recognized that 
algebra is on a higher level of abstraction than arithmetic. This 
advance in generality was one of the most important steps ever 
taken in mathematics. The complete divorce of algebra and 
arithmetic was consummated only in the nineteenth century, 
when the postulational method freed the symbols of algebra from 
any necessary arithmetical connotation. 

Improving on the devices of his European predecessors, Vieta 
gave a uniform method for the numerical solution of algebraic 
equations. Its nature is sufficiently recalled here by noting that 
it was essentially the same as Newton’s (1669) given in text- 



books. Although Vieta’s method has been displaced by others, its 
historical significance is of more than antiquarian interest. The 
method applies to transcendental equations as readily as to 
algebraic when combined with expansions to a few terms by 
Taylor’s or Maclaurin’s series. 

An algebraic equation of degree 45 which Vieta attacked in 
reply to a challenge indicates the quality of his work in trigonom- 
etry. Consistently seeking the generality underlying particulars, 
Vieta had found how to express sin nd (n a positive integer) as a 
polynomial in sin 0, cos 0. He saw at once that the formidable 
equation of his rival had been manufactured from an equivalent 
of dividing the circumference of the unit circle into 45 equal 
parts. But for his lack of negative numbers, Vieta would have 
found all 45 roots instead of the 23 he did. More important 
than this spectacular feat rvas Vieta’s suggestion that cubics 
can be solved trigonometrically. Indicative of the general 
haziness of algebra in the time of Vieta, his partial failure 
underlines the fact that even toward the close of the sixteenth 
century there was no clear conception of what is meant by the 
roots of an algebraic equation. Once more the obscurity arose 
from an incomplete understanding of the number system of 

As evidence of Vieta’s modern or Archimedean freedom, his 
application of both algebra and trigonometry to geometrical 
problems may be instanced, particularly in the use of algebra to 
replace geometric constructions wherever feasible. That he found 
nothing new in geometry of any lasting importance is immate- 
rial; it was the boldness of his thought that mattered. But, 
systematic and general within its necessary limitations as this 
algebraized geometry was, it would be giving it infinitely more 
than its due to call it a precursor of analytic geometry in any 
but the strictly chronological sense. It nowhere even hints at the 
spirit of analytic geometry which made analysis and geometry 
complementary aspects of one mathematical discipline. 

Victa’s principal advance in trigonometry was his systematic 
application of algebra. In both plane and spherical trigonometry he 
worked freely with all six of the usual functions, and in the former 
obtained many of the fundamental identities algebraically. 

With Vieta, elementary (non-analytic) trigonometry was 
practically completed except on the computational side. All 
computation was greatly simplified early in the seventeenth 
century by the invention (1614) of logarithms. In p re-logarithmic 


computation Vieta extended (1579) the tables (1551) by G. J. 
Rhaeticus (1514-1576, German), giving the values to seven 
places of all six functions for every second of arc, instead of for 
every ten seconds as in Rhaeticus. The deliberate separation of 
trigonometry from astronomy is usually credited to Regiomon- 
tanus (Johannes Muller, 1436-1476, German) in his systematic 
De triangulis of 1464. 

Elementary algebra at the close of the sixteenth century had 
still to receive many perfections, especially in notation, before 
it became the simple routine of our textbooks. But by 1600 the 
straight path for all future development had been clearly indi- 
cated. The way which Vieta had pointed out was followed with 
brilliant success by a host of workers, most of whom made 
incidental improvements in algebraic notation and technique 
while developing their major interests in the new mathematics. 
There was not another first-rank algebraist after Vieta till the 
eighteenth century, 5 when Lagrange sought and found deeper 

To conclude this account of Vieta, we quote the opinion of a 
first-rate mathematician and mathematical historian on Vieta’s 
place in the history of mathematics. Writing in 1843, De Morgan 
expressed himself as follows. 

Vieta is a name to which it matters little that we have not dwelt on several 
points which would have made a character for a less person, such as his com- 
pletion of the cases of solution of right-angled spherical triangles, his expres- 
sions for the approximate quadrature of the circle, his arithmetical extensions 
of the same approximations, and so on. The two great pedestals on which his 
fame rests are his improvements in the form of algebra, which he first made a 
purely symbolical science, and showed to be capable of wide and easy applica- 
tion in ordinary hands; his application of his new algebra to the extension of 
trigonometry, in which he first discovered the important relations of multiple 
angles; and his extension of the antient rules for division and extraction of the 
square and cube roots to the exegetic process for the solution of all equations. 
. . . If a Persian or an Hindu, instructed in the modern European algebra 
were to ask, “Who, of all individual men, made the step which most distinctly 
marks the separation of the science which you now return to us from that 
which we delivered to you by the hands of Mohammed Ben Musa [Al-Kho- 
warizmi]?” the answer must be — Vieta. . . . When will the writer who 
asserts that Cardan was substantially in possession of Vieta’s algebra attempt 
to substantiate his assertion by putting so much as half a page of the former 
side by side with one of the latter ? 

The development of symbolism 

The importance of an easily manipulated symbolism, as 
implied by De Morgan, is that it enables those who are not 



great mathematicians in their generation to do without effort 
mathematics which would have baffled the greatest of their 
predecessors. The formulas in an engineers’ handbook, for 
instance, if transposed into concise verbal equivalents, with a 
liberal use of abbreviations and conventional signs for the most 
frequently occurring words, might be intelligible to an Ar- 
chimedes; to the average engineer they would probably be 
exasperating gibberish. And the prospect of having to combine 
several such verbalized formulas, in the hope of gaining useful 
information, might discourage even a modern Archimedes. 
In mathematics itself, as distinguished from its applications, 
the situation is the same. Unless elementary algebra had become 
“a purely symbolical science” by the end of the sixteenth 
century, it seems unlikely that analytic geometry, the differential 
and integral calculus, the theory of probability, the theory of 
numbers, and dynamics could have taken root and flourished 
as they did in the seventeenth century. As modern mathe- 
matics stems from these creations of Descartes, Newton and 
Leibniz, Pascal, Fermat, and Galileo, it may not be too much to 
claim that the perfection of algebraic symbolism was a major 
contributor to the unprecedented speed with which mathe- 
matics developed after the publication of Descartes’ geometry 
in 1637. It is therefore of interest in following the evolution of 
mathematics to review the principal stages by which elementary 
algebraic symbolism reached its present maturity and to note 
how the lack of an effective symbolism hampered the progress 
of mathematics in some of its more productive periods. Two 
general observations may help to clarify the somewhat confused 
historical record. 

In his analysis of Greek algebra (1842), G. H. F. Ncssclmann 
(German) noted three historical phases of algebra, to which 
he gave the suggestive names rhetorical, syncopated, and 
symbolic. In the earliest phase, the rhetorical, the entire state- 
ment and solution of an algebraic problem were wholly verbal. 
It is not exactly clear why the outcome should be called algebra 
at all, unless it be that similar problems and their solutions 
reappeared in a later phase scantily clothed in at least the sug- 
gestion of a symbolism. The middle phase, the syncopated, was 
distinguished from the first only by the substitution of abbrevia- 
tions for the more frequently occurring concepts and operations. 
Syncopated ‘algebra’ was thus an early instance of the quarter- 
truth that “mathematics is a shorthand.” If algebra were 


nothing more than a shorthand, its contribution to the rudi- 
ments of mathematical thought would not be very impressive. 
The third phase, the symbolic, presents algebra as fully sym- 
bolized with respect to both its operations and its concepts. 
It also does much more than this. 

Symbolic algebra replaces verbalized algebraic processes, 
which cost the practitioners of rhetorical and syncopated algebra 
much patient thought, by symbolic procedures summarizing 
chains of verbal reasoning in readily apprehended rules requiring 
only passive attention. The experience gained through centuries 
of laborious trial is condensed in mechanical processes which 
can be applied and manipulated with a minimum of thinking. 
If such manual dexterity as almost suffices for competence in 
solving linear equations, say, is condemned — as it frequently is — 
for its all but negligible educational value, it has had the merit 
of liberating the higher faculties of mathematicians to attack 
problems more difficult than any that taxed the devious 
ingenuity of the Greeks, the Indians, the Moslems, and the alge- 
braists of the early Renaissance. In even elementary mathe- 
matics there is still opportunity enough for invigorating and 
profitable mental exercise. Finally, symbolic reasoning, as in 
the current phase of algebra, has suggested extensive generaliza- 
tions and economical unifications. A typical example was the 
introduction (1655) of negative and fractional rational ex- 
ponents, culminating about two centuries later in arbitrary 
complex exponents with a satisfactory theory to justify their 

The second general remark regarding the evolution of mathe- 
matical symbolism is implicit in the recognition of the three 
phases of algebra. As algebra progressed, a multitude of in- 
dividual names for members of what came to be recognized 
as one inclusive class were abandoned in favor of a uniform 
terminology significant for all members of the class. Further, in 
several instances, uniformity was possible only because the 
several members of the class were unified by some underlying 
property, usually simple when at last uncovered, of the rational 
numbers. When such was the case, an appropriate numerical 
character was imposed on the whole class, and an algoristic 
symbolism, amenable to the operations of rational arithmetic, 
brought the algebraically important characteristic of the class 
within the grasp of all but involuntary manipulative skill. 
For example, when it was finally perceived, after centuries of 



overlooking the elusive fact which now seems obvious, that the 
powers x, x 2 , a* 3 , A' 4 , a* 6 , a 6 , . . . arc unified with respect to their 
exponents 1, 2, 3, 4, 5, 6, , and that multiplication of 

powers of the unknown is effected by addition of exponents, 
an incredible mass of confusing terminology and inefficient 
rules was swept into the past, and with it, an equal or greater 
mass of tortuous thinking. Similar syntheses, again originating 
in some concealed but gradually perceived property of the 
rational numbers, accompanied the growth of the number 
concept. To cite a simple instance, the equations ax 2 + bx — c, 
ax 2 = bx + c, with a, b, c positive rational numbers, presented 
two distinct problems to algebraists before negative rational 
numbers were handled correctly and with (unjustified) con- 
fidence. The use of negatives reduced the solution of the two 
equations to that of the single equation ax 2 -f- bx -f* c — 0, 
with a, b, c rational numbers. 

In passing, it is strange to find the two special quadratics 
treated independently in textbooks of less than a century ago. 
But perhaps this is not remarkable when we remember that 
Gauss consistently wrote a -2 as xx, for the curiously unmathe- 
matical reason that neither is more wasteful of space than the 
other. For ourselves, we still call x 2 and a * 3 the square and the 
cube of a*, possibly for easy diction, or perhaps because some of 
the ancients were mathematically fluent in the limpid jargon 
of areas and volumes. But there arc already hints that squares 
and cubes may obsolcscc from the vocabulary of algebra before 
many more centuries have passed. In the meantime, no tyro 
in algebra need be seriously discommoded by his inability to 
sec immediately what squares and cubes have to do with a, 
provided only we do not afflict him with the corresponding 
names for a' 4 , a -5 , a 5 , . . . from the geometry of hypcrspacc, 
as we should do if wc arc to foster linguistic purity. But, as was 
discovered by too hasty penitents in the Middle Ages, even 
purity may sometimes cost too much for comfort. 

The absence of symbolism in Babylonian algebra, already 
noted, poses the problem of what is to be recognized as algebra 
in the rhetorical and syncopated phases. Since it seems to be 
agreed among historians of mathematics that the Babylonians, 
the Egyptians, Diophantus, and the more rhetorical Moslems 
actually practiced a more or less rudimentary algebra, it is 
clear that the absence or presence of symbolism is not the his- 
torical criterion. More than a mere matter of words is involved. 


From the frequently decisive impacts of symbolism on the gen- 
eral development of mathematical reasoning, it would seem 
that mathematics itself and not the pedantries of terminology is 
the important issue. An older conception of algebra than that 
now universal identified algebra with the solution of equations. 
If this antique be admitted, both the rhetorical and the synco- 
pated phases are accepted as algebra without further qualifica- 
tion, and the entire historical development of the subject down 
to the beginning of the nineteenth century acquires a deceptive 
unity and a specious coherence. A somewhat similar conception 
of algebra indicates the use of unknowns, whether verbalized 
or symbolized, as the historical clue to be followed. This, 
however, comprises too much, as it includes such geometrical 
problems as the construction of a circle to satisfy prescribed 
conditions. To narrow the scope of the ‘unknown’ sufficiently, it 
may be restricted to the domain of numbers — a restriction which 
lost its stringency with the invention of analytic geometry. In 
an obvious sense all mathematics is a quest for the unknown. 
In addition to affording a clue to the development of symbolism, 
this inclusive definition has the signal advantage over its too 
numerous competitors of permitting the algebraists or the anal- 
ysts to claim all mathematics as their province, as already done 
by some of the geometers. 

In spite of all objections, the available data seem to show 
that for following the historical development of symbolism, 
either equations or unknowns offer a convenient directive. 
Again, the mere existence of equations all through the protracted 
evolution also foreshadowed a most important aspect of mathe- 
matical thought, which dominated much of the work of the 
recent period beginning in 1801 , that of mathematics as a study 
of relations. In the earliest stages, the only relation considered 
was equality, and it required about three thousand years for 
this ubiquitous concept to reach full symbolic representation. 
This may serve as a typical example of the slowness with which 
the commonest paraphernalia of current mathematics evolved. 

Operations seem to have been symbolized more readily than 
relations. If this is a correct statement of the facts, it accords 
with the order of increasing abstractness. But success in one 
department apparently did not appreciably stimulate inventive- 
ness in another, and until the recent period the development of 
symbolism proceeded haphazardly. 

In modern mathematics the creation of an efficient notation 


12 7 

may sometimes have been accidental, but usually it ivas the 
outcome of conscious effort. An example of the first is the nota- 

tion j (not a/b) for fractions, an invention whose full value 
may not have been appreciated by its author. Possibly the most 

striking instance of the second is Leibniz’ -y- (not dy/dx r) for 

the derivative of y with respect to x. Nobody in the history of 
mathematics was more sensitive than Leibniz to the potentiali- 
ties of a rationally devised symbolism, and nobody gave the 
‘philosophy’ of mathematical notation more painstaking 
thought than he. More recently, the notations a/b for fractions 
and dy/dx for derivatives in some respects have illustrated 
progress in reverse. Centuries of easy habit were discarded 
to accommodate the incompetence of printers — the modern 
reinventor of a/b gave substantially this reason for his departure 
from custom. In our own century the incompetence of machines 
contributed in a similar manner to the delinquence of happily 
conceived notations, for example that of the tensor calculus, 
until printers learned that it pays to hire competent engineers 
to revise their machinery. This is one of the few instances where 
the economic motive has reacted simply and directly with the 
ideal of mathematical clarity to the benefit of both. In the 

matter of -y the motivation was partly a humane desire to mod- 

erate the repulsivcncss of vulgar fractions to young children. 

A small sample of the many notations for powers of the 
unknown will suffice to suggest the progression from rhetorical 
to symbolic algebra. Ahmcs (seventeenth century b.c.) used a 
word, variously translated as ‘heap,’ ‘amount,’ ‘mass,’ to 
denote the unknown. Diophantus (third century a.d.) used a 
shorthand for the successive powers of the unknown: x z was 
the ‘power,’ x 3 the ‘cube,’ and ‘power,’ ‘cube’ were denoted 
by (what were probably) abbreviations of the corresponding 
Greek words. Say these abbreviations were P, C; then PP, PC, 
CC, denoted the fourth, fifth, sixth powers of the unknown, and 
so on. Here evidently there is a rationale behind the syncopation. 
Such a notation was but ill adapted to the simultaneous repre- 
sentation of several unknowns. Traces of operational symbolism 
also arc attributed to Diophantus. Addition was indicated by 
juxtaposition, subtraction by a special symbol whose genesis is 


still in dispute — it may have been the first letter of a Greek 
word, or a genuine operational symbol in the sense that it was 
not derived from an abbreviation. If the latter, it was a sig- 
nificant step toward symbolic algebra. But Diophantus did not 
rise to relational symbolism, using the first two letters of the 
Greek word for ‘equality’ to denote ‘is equal to.’ It is doubted by 
some scholars whether Diophantus made any use of symbolism, 
his claim to having done so resting on a manuscript of his 
arithmetic written about a thousand years after his death. 

Both the Indians and the Moslems followed Diophantus in 
what may be described as additive juxtaposition to denote 
successive powers of the unknown. Aryabhatta (fifth to sixth 
centuries a . d .) abbreviated the unknown to ya, its second, third, 
fourth, sixth powers to va, glia, va va, va glia , and so on. He 
also provided for several unknowns; ya (the first unknown), ka, 
ni, pi (the second, third, fourth unknowns) the abbreviations 
being those of the color names black, blue, yellow. Operations 
were indicated after the operands by the words ghata, bha, indi- 
cating addition, multiplication. Thus xy, is ya ka bha, where x, y 
are the unknowns. The substitute for a sign of equality was 
adequate, one of two equals being written under the other. The 
word for ‘root’ was mula. Combined with other words, as 
varga mula, ghana mula for ‘square root,’ ‘cube root,’ mula was 
hardly so much a mathematical symbol as a common noun. A 
closer approach to symbolism was the indication of the negative 
of a number by writing a dot or a small circle above the number. 
Of the Moslems, Al-Khowarazmi (first half of the ninth century), 
used jidir (root) for the unknown, and vial (power) for its 
square. Al-Karkhi (early eleventh century), with kab for the 
cube, composed the fourth, fifth, sixth, seventh, . . . powers 
by juxtaposition, as mal mal, mal kab, kab kab, mal mal kab. The 
Moslems generally followed Diophantus in simplifying equations 
by combining like terms. Both appear to have been led astray 
by this natural simplification. The significant classification of 
equations in one unknown is not according to number of terms, 
but by degree. However, the later Moslems, in spite of their 
ineffectual protosymbolism, recognized that the next problem 
after the quadratic in equations was the cubic, by no means an 
easy recognition from their point of view. 

Late in the fifteenth century, the Moslems approached a 
purely symbolic representation of an operation in writing only 
the first letter of the Arabic for ‘root’ above a number to indicate 



square root. This might be considered an intermediate stage 
between syncopation and a matured symbolism in which opera- 
tions are designated by specially devised signs whose verbal 
origin, if any, is no longer recognizable. An example of the last 
is the current sign for equality. Unless it were definitely known 
that Recorde invented this sign ( Whetstone of wiite, 1557), it 
might well be mistaken for a degenerated form of a word in 
medieval shorthand. But Recorde denoted equality by = be- 
cause, it seemed to him, no two things could be “moare equalle” 
than “a paire of paralleles” — which is reminiscent of the 
remark that “William and John,” twins, “are very much alike, 
especially William.” But Recorde had the true symbolist’s 
instinct for the ultimate perfection, by whatever conceit he 
chose to propitiate his syncopating contemporaries. The 
Egyptians had used the hieratic form of their hieroglyph for 
‘equality’; the Greeks, the first two letters of their word, the 
Moslems, the last letter of their word, till they reverted to 
total verbalism and wrote out ‘equality’ in full. It remained for 
Recorde to do the right thing. 

In equations, the passage through the three phases was 
similar to the evolution of the progressively more algoristic 
notation for powers, which reached a climax in Wallis’ (1655) 
.V", A a/n for l/.v n , yjx. Greek equations were partly rhetorical, 
partly syncopated, with little that would now be recognized 
as algebraic symbolism. Al-Khowarizmi’s equations were purely 
rhetorical; a Latin translation of one is “census et quinque radices 
cquantur viginti quatuor or “the square of the unknown ( census ) 
and five unknowns ( radices ) are equal to twenty four,” that is 
x 5 -f- 5x — 24. The Europeans of the sixteenth and seventeenth 
centuries gradually approached full symbolism in the writing 
of equations, as seen in the following specimens. Cardan (1545) 
wrote + 6x = 20 as “ cubus p 6 rebus aequalis 20,” in which 
there is nothing to indicate that cubus and rebus are powers 
(third, first) of the same unknown. The p is ‘plus.’ Vieta, with 
C, Q, N for ‘cube,’ ‘square,’ ‘number’ or ‘unknown,’ also left 
it to be inferred that these terms refer to one unknown in 
IC — 8£? + 16iV aequ. 40. At last Descartes (1637) settled the 
matter (except that he missed x- for xx) by writing .v, xx, x s , 
x \ **, ... for Vieta’s A 7 , Q, C, QQ, QC , . . . , putting all 
positive integral powers on the uniform notational basis familiar 
today. It all seemed so simple when it was finally done after 
centuries of effort. 


Volumes might be (and have been) written about the evolu- 
tion of mathematical symbolism. Probably almost anyone 
leafing through these will agree that lack of appropriate sym- 
bolism constrained the Greek arithmeticians and algebraists to 
consider special cases of what might have been their problems, 
and prevented the Indians and the Moslems from producing an 
elementary algebra within the capacities of ordinary adolescents. 


The Beginning of Modern 

Historical sketches may sometimes decoy us into artificial 
divisions of human progress by centuries or half-centuries de- 
marked by precise dates. Having just passed through one such 
critical fifty years, we may well suspect that another is more 
imaginary than real. Be this as it may, the half-century from 
1637 to 1687 is universally recognized as the fountainhead of 
modern mathematics. The first date marks the publication of 
Descartes’ Geometric, the second, that of Newton’s Principia. 1 

From this prolific period, as from the Greek golden age, we 
shall select only those contributions which overtop a multitude 
of interesting details in their significance for the development of 
all mathematics. Some of the items omitted will be noted in 
later chapters, where they may be naturally included without 
interrupting the continuity of the main current. Thus, in this 
period infinite series advanced notably, but were of minor impor- 
tance compared with the calculus. Again, the contribution of 
Leibniz to symbolic logic can be best described in the light of 
modern work, and will be noted in the final chapter. It may be 
remarked once for all that mathematics overshadows its creators; 
that we arc primarily interested in mathematics; and that each 
of the men cited did far more than the few items described here, 
but that much of what is omitted has for long been of only 
antiquarian interest. 



As some of the men whose work is to be reviewed were more 
directly responsible than others for the creation of modern 
mathematics, they will be given more extended notices than 
might be justified on purely impersonal grounds. Like Pythag- 
oras, they too will doubtless vanish as personalities and live 
only in the body of mathematics as the centuries slip away; but 
at present they are close enough to us to be more than names 
attached to mathematical abstractions. 

These outstanding originators of modern mathematics were 
not merely half a dozen eminent men in a crowd; they towered 
above the majority of those who preceded or came after them. 
Conspicuous eminence in mathematics was harder to achieve 
after these men had lived, simply because by the power of their 
methods they had quite suddenly raised the whole level of attain- 
able mathematics. Geometers, for instance, were no longer 
condemned to crawl among the five conics and a handful of 
simple higher plane curves after Descartes had given them wings. 
It is arguable that even the most original of these men was 
indebted to the very humblest of those who preceded him. But 
their incomparable superiority in generality of outlook almost 
inclines us to regard all of them rather as sudden mutations 
touched off into explosive activity by accidents of their environ- 
ment, than as orderly end products of a creeping evolution. 

Five major advances 

Modern mathematics originated in five major advances of 
the seventeenth century: the analytic geometry of Fermat 
(1629) and Descartes (1637); the differential and integral cal- 
culus of Newton (1666, 1684) and Leibniz (1673, 1675); the 
combinatorial analysis (1654), particularly the mathematical 
theory of probability, of Fermat and Pascal; the higher arith- 
metic (c. 1630-65) of Fermat; the dynamics of Galileo (1591, 
1612) and Newton (1666, 1684), and the universal gravitation 
(1666, 1684-7) of Newton. 

With these five, two further departures in new directions may 
be cited for their influence on subsequent advances: the syn- 
thetic projective geometry (1636-9) of Desargues and Pascal; 
the beginning of symbolic logic (1665-90) by Leibniz. 

Throughout the first half of the century, reactionary hos- 
tility to science continued its losing fight, reaching its futile 
climax in the condemnation of Galileo by the Inquisition in 
1633, only four years before Descartes, safe in Holland, per- 


mitted his masterpiece to be printed. Intolerance mas partly 
offset by the scientific societies founded during this period or 
shortly after. Only the three most influential need be mentioned. 
The Royal Society of London was incorporated in 1662; Newton 
was its president from 1703 to 1727. The French Academy of 
Sciences (Academie des Sciences, Paris) crystallized in 1666 
from the informal meetings of a group of savants, some of whom, 
including Mersenne, Descartes, and Mydorge, were primarily 
mathematicians. The Berlin Academy (Societat der Wissen- 
schaften) was founded in 1700 at the instigation of Leibniz. He 
was its first president. The Paris and Berlin Academies have 
consistently been more cordial than the Royal Society to pure 

The importance of these and other academies for the ad- 
vancement of science during the seventeenth and eighteenth 
centuries cannot be overestimated. Together, they did far more 
than the universities for science, one of their chief functions 
being the publication of research by their members. Even more 
important than this was the living example each of the scientific 
societies afforded of a nucleus of intelligent, influential men in a 
society still cowed by religious bigotry and scholarly intolerance. 
By the end of the seventeenth century, science had grown too 
sturdy for indiscriminate attack; and the forces of reaction, 
fighting among themselves, lacked the wit to combine against 
their common enemy. 

A remarkable feature of the rapid development in mathe- 
matics was that the continuous and the discrete divisions ad- 
vanced simultaneously. The advance in the continuous might 
have been expected; the other has the appearance of an accident. 
Neither more or less trivial arithmetic of permutations and 
combinations, nor unsystematized observations on games of 
chance, offer a sufficient explanation of the sudden and complete 
emergence of the fundamental principles of the theory of 

The most prolific of all the new acquisitions was the calculus; 
for when geometry became analytic, it derived most of its life 
from the analysis of functions continuous except at isolated 
singularities. There was thus provided an infinite store of curves 
and surfaces on which geometers might draw, and to which they 
applied the methods of the calculus to discover and investigate 
exceptional points, such as cusps and inflections, not intuitively 
evident from the equations. 


It is not surprising that for more than a century after it 
became public property, the calculus and its applications to 
geometry, dynamical astronomy, and mechanics attracted all 
but a few of the ablest men, to the comparative neglect of 
combinatorial analysis, the theory of numbers, algebra (except 
improvements in notation and Descartes’ work in equations), 
symbolic logic, and projective geometry. For more than twenty 
centuries, geometry and astronomy had dominated mathe- 
matical tradition in the works of the masters. Now here at last 
was the universal solvent for all the intractabilities of classical 
geometry and astronomy, and the philosopher’s stone that 
changed everything it touched to gold. Difficulties that would 
have baffled Archimedes were easily overcome by men not 
worthy to strew the sand in which he traced his diagrams. 
Leibniz did not exaggerate when (1691) he boasted that “My 
new calculus [and Newton’s] . . . offers truths by a kind of 
analysis, and without any effort of the imagination — which 
often succeeds only by accident — ; and it gives us all the advan- 
tages over Archimedes that Vieta and Descartes have given us 
over Apollonius.” 

The calculus of Newton and Leibniz at last provided the 
long-sought method for investigating continuity in all of its 
manifestations, whether in the sciences or in pure mathematics. 
All continuous change, as in dynamics or in the flow of heat and 
electricity, is at present attackable mathematically only by the 
calculus and its modern developments. The most important 
equations of mechanics, astronomy, and the physical sciences 
are differential and integral equations, both outgrowths of the 
seventeenth-century calculus. In pure mathematics, the calculus 
at one sweep revealed unimagined continents to be explored and 
reduced to order, as in the creation of new functions to satisfy 
differential equations with or without prescribed initial condi- 
tions. One of the simplest of all such equations, dy = f{x)dx, in a 
sense defines the integral calculus; and the corresponding 
integral, jf(x)dx, alone suggests an endless variety of functions 
according to the form of f(x). 

In the discrete division, continuity is of only secondary 
importance. Primarily, combinatorial analysis is concerned with 
the relations between subclasses of a given class of discrete 
objects, for example with the interrelations of the permutations 
and combinations of the members of a given countable class. 
Fermat’s and Pascal’s work of 1654 on probability lifted com- 


binatorial analysis from the domain of mathematical recreations 
into that of severely practical mathematics; and only about 
fifty years elapsed between the creation of the mathematical 
theory of probability and the calculation of mortality tables by 
its use. In modern combinatorial analysis the calculus is indis- 
pensable in obtaining usable approximations to formulas beyond 
practicable exact computation. 1 

The other great advance in the discrete division, Fermat’s 
creation of the modern higher arithmetic, was for long restricted 
to the study of relations between subclasses of the class of all 
rational integers. Since about 1850, numerous arithmeticians 
have extended the classical theory of Fermat and his successors 
to vastly wider classes of integers. The contribution to all 
mathematics of the higher arithmetic has been indirect, in the 
invention of new techniques, particularly in modern higher 
algebra and to a lesser extent in analysis, primarily for applica- 
tion to problems concerning the rational integers. Conversely, 
extensive tracts of the modern theory of numbers would not exist 
had not analysis made them possible. 

The careers of synthetic projective geometry and symbolic 
logic afford an interesting contrast in mathematical obsolescence 
and survival. Both will be noted in subsequent chapters; here we 
remark only the striking difference between their fate and the 
uniform prosperity of the other creations of the seventeenth 
century. After its invention by Dcsargucs and Pascal, synthetic 
projective geometry languished till the early nineteenth century, 
when it became extremely popular among geometers with a 
distaste for analysis. Leibniz’ dream of a mathematical science of 
deduction lay dormant till the mid-nineteenth century, and 
even then it appealed to but very few, atthough Leibniz had 
foreseen the importance of symbolic logic for all mathematics, 
and had himself made notable progress toward an algebra of 
classes. Only in the second decade of the twentieth century did 
mathematical logic become a major division of mathematics. 
Concurrently, synthetic projective geometry was receding defi- 
nitely into the past with the reluctant admission that an essen- 
tially Greek technique, even when revitalized, is hopelessly 
impotent in competition with the analytic methods of Descartes 
and his successors. 

From all this it is clear that after the period of Archimedes, 
Euclid, and Apollonius, that of Descartes, Fermat, Newton, and 
Leibniz is the second great age of mathematics. If the funda- 


mental distinction between the old and the new can be suggested 
in a word, it may be said that the spirit of the old was synthesis 
that of the new, analysis. 

* Anticipations’ 

Before proceeding to the individual advances, we must dis- 
pose of a purely historical matter which will not be further 
discussed. It concerns numerous aborted or sterile ideas which 
have not passed into living mathematics. 

Behind each of the major advances of the seventeenth cen- 
tury were many short steps in the general direction of each, and 
some of these partial advances all but reached their unperceived 
goals. At least that is what we might be tempted to imagine now. 
Looking back on these efforts, some may be inclined in the 
generosity of their hearts to believe that without these halted 
steps final success would have been long delayed or unattained. 
In the specific instances of analytic geometry and the calculus, an 
examination of the mathematics — not the sentiments — involved 
has convinced a majority of professionals that the alleged 
anticipations are illusory. Especially is this the opinion of men 
who themselves create mathematics and who know from dis- 
concerting experience that hindsight sees much to which fore- 
sight was blind. 

In retrospect we can trace the evolution of analytic geometry, 
for example, back to Hipparchus, or even to the ancient Egyp- 
tians. Like every astronomer who has recorded the positions of 
the planets, Hipparchus used coordinates, in particular latitude 
and longitude. But the use of coordinates entitles nobody to 
priority in the invention of analytic geometry; nor does even an 
extensive use of graphs. As any intelligent beginner who has 
understood the first three weeks of a course in analytic geometry 
knows, analytic geometry and the use of coordinates in the plot- 
ting of graphs are a universe apart. Only in the sense that they 
preceded analytic geometry are such comparatively childish 
activities anticipations of that geometry. This also is the judg- 
ment of a majority of professional geometers, who probably 
are as competent as anyone in this matter. 

Wc must refer the reader elsewhere 3 for a detailed evaluation 
and rejection of the romantic claims that several early mathe- 
maticians, and in particular Apollonius, Nicole Oresme (four- 
teenth century), and Kepler, ‘anticipated’ Descartes and Fermat 
in their independent invention of analytic geometry. To preserve 


the balance, ana to exhibit 2 clean-cut instance of the absolute 
zero to which so many scholarly differences of opinion in the 
history of mathematics add up, we cite another evaluation of 
these ‘anticipations’ in which exactly the opposite conclusion 
is ably upheld.' 4 Dozens more on either side might easily be 
mentioned; the two selected will suffice to orient whoever may 
be interested, and to start him on his own critical evaluations. 

In each of the major advances of the seventeenth century 
some definite step led from confusion to 2 new method. Thus 
Newton himself states what gave him a hint for the differential 
calculus: “Fermat’s way of drawing tangents.” 

But there is one ‘anticipator’ of the calculus, B. Cavalieri 
(1598-1647, Italian), who merits more than a passing citation for 
the lasting mischief his ‘anticipation’ has done. Cavalieri's 
method of indivisibles has endured, to the distraction of hun- 
dreds of teachers of the elementary calculus who must extirpate 
heretical notions of infinitesimals from their students’ minds. 

In the United States, much of this elementary confusion can 
be traced to a generation of college teachers who were thoroughly 
indoctrinated in their school course in solid geometry with 
Cavalieri’s method of indivisibles. Their school geometries 
contained a seductive section on what some textbook writers 
called Cavalieri bodies: and these indivisibly-divisible non- 
entities were used, among other absurdities, to inculcate a disas- 
trously nonsensical account of mensuration in three dimensions. 

Cavalieri did not anticipate the calculus: he committed the 
unpardonable sin against it. But for his indivisibles and their 
absorption by scores of otherwise rational men who were to 
become college teachers, the common delusion that an infini- 
tesimal is a ‘little zero’ would have been extinct two generations 

The historical appeal of Cavalieri’s indivisibles is undeniable, 
and that, perhaps, is why some historians palliate their flagrant 
offenses. They were inspired by the scholastic lucubrations of 
Thomas Bradwardine, Archbishop o? Canterbury (thirteenth 
century), and the submathcmatical analysis of Thomas Aquinas. 
As Cavalieri never defines his indivisibles explicitly, it is open 
to his apologists to read into them anything they know should be 
there but is not. But if his mystical exposition (1635) means 
anything at all. Cavalieri regarded a line as being composed of 
points like a string of countable but dimensionless beads, a 
surface as made un simiiarlv of lines without breadth, and a solid 


as a stack of surfaces without ultimate thickness. These are the 
very notions which a conscientious teacher will purge out of his 
students if it takes four years. 

A historical argument in favor of these indivisibles is that 
Leibniz was acquainted with them. But even this does not make 
Cavalieri an anticipator of the calculus. As will appear shortly, 
Newton clearly recognized the untenability of indivisibles; and 
although he did not fully succeed in clearing up his own diffi- 
culties, he did not mistake nonsense for sound reasoning. That 
is the fundamental distinction between one who imagined the 
calculus and one who did not. Contrary estimates of Cavalieri’s 
work are readily available. 

Descartes , Fermat , and analytic geometry 

Rene Descartes (1596-1650, French) is more widely known as 
a philosopher than as a mathematician, although his philosophy 
has been controverted while his mathematics has not. 

Descartes’ family was of the lesser French nobility. His 
mother died shortly after her son’s birth, but an unusually 
humane father and a capable nurse made up for this loss. After a 
broad education in the humanities at the Jesuit college of La 
Fleche, Descartes lived for two years in Paris, where he studied 
mathematics by himself, before joining Prince Maurice of Orange 
at Breda as a gentleman officer in 1617. In 1621, Descartes aban- 
doned his military career, partly because he had seen enough 
service both active and passive, partly because, as he declared, 
three dreams on the night of November 10, 1619, suggested the 
germs of his philosophy and analytic geometry. 

Much of the remainder of his life was spent in Holland, where 
he was safer from possible religious persecution than he would 
have been in France. These were his productive years; and in 
spite of his desire'tfor tranquillity, he could not conceal the 
greatness of his thowht. Rumors of what he was thinking were 
discussed wherever cshers with minds akin to his own dared to 
think. Largely througVf he efforts of F ather M. Mersenne (1588— 
1648, French) of Pariycwho acted as intermediary between the 
French intellectuals ail«<the justly cautious Descartes, his fame 
spread over all Europei’t 

In 1637 Descartes published the work on which his greatness 
as a mathematician rests, the Discours de la methode four Men 
conduire sa raison et chercher la verite dans les sciences , the third 


and last appendix of which, La gcomclrie , contains his subversive 

The closing months of his life were spent as tutor to the young 
and headstrong Queen Christina of Sweden. The rigors of a 
Stockholm winter and the inconsiderate demands of his royal 
pupil caused his death. In accordance with the ideals of his age, 
when experimental science was first seriously challenging arro- 
gant speculation, Descartes set greater store by his philosophy 
than his mathematics. But he fully appreciated the power of his 
new method in geometry. In a letter of 1637 to Mersenne, after 
saying “I do not enjoy speaking in praise of myself,” Descartes 
continues: "... what I have given in the second book on the 
nature and properties of curved lines, and the method of examin- 
ing them, is, it seems to me, as far beyond the treatment in the 
ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c 
of children.” 6 

The famous appendix 6 on geometry consists of three books, of 
which the second is the most important. The third is devoted 
mostly to algebra. It will suffice to restate in current terminology 
the essential features of Descartes’ advance.® 

A plane curve is defined by some specific property which 
holds for each and every point on the curve. For example, a 
circle is the plane locus of a point whose distance from a fixed 
point is constant. Any point on a curve is uniquely determined by 
its coordinates x and y; and an equation /(.v, y) = 0 between the 
coordinates completely represents the curve when the specific 
geometric property defining the curve is translated into a rela- 
tion, denoted by the function /, between the coordinates x, y 
of the particular-general point on the curve. 

There is thus established a one-one correspondence between 
plane curves and equations in two variables x } y: for each curve 
there is a definite equation /(.v, y) = 0, and for each equation 
/(.V, y) = 0 there is a definite curve. 

Further, there is a similar correspondence between the alge- 
braic and analytic properties of the equation /(.v, y) = 0 and the 
geometric properties of the curve. Geometry is thus reduced to 
algebra and analysis. 

Conversely, analysis may be spoken in the language of 
geometry, and this has been a fecund source of progress in 
analysis and mathematical physics. 

The implications of Descartes’ analytic reformulation of 
geometry are obvious. Not only did the new method make possi- 


ble a systematic investigation of known curves, but, what is of 
infinitely deeper significance, it potentially created a whole 
universe of geometric forms beyond conception by the synthetic 

Descartes also saw that his method applies equally well to 
surfaces, the correspondence here being between surfaces defined 
geometrically and equations in three variables. But he did not 
develop this. With the extension to surfaces, there was no 
reason why geometry should stop with equations in three vari- 
ables; and the generalization to systems of equations in any finite 
number of variables was readily made in the nineteenth century. 
Finally, in the twentieth century, the farthest extension possible 
in this direction led to spaces of a non-denumerable infinity of 
dimensions. The last are not mere fantasies of the mathematical 
imagination; they are extremely useful frameworks for much of 
the intricate analysis of modern physics. The path from 
Descartes to the creators of higher space is straight and clear; 
the remarkable thing is that it was not traveled earlier than it was. 

Another direct road from Descartes to the present may be 
noted in passing. The formula (#1 — # 2 ) 2 + (yi — y 2 ) 2 for the 
square of the distance between any points (xi, yi), (x 2 , y 2 ) in a 
plane (surface of zero curvature) suggested the corresponding 
formulas in differential geometry for the square of the line 
element joining neighboring points in any space, flat or curved, 
of any number of dimensions, as quadratic differential forms. 
The germ of this long evolution was the Pythagorean theorem. 

In details, Descartes’ presentation differs from that now 
current. Thus, he used only an ar-axis and did not refer to a y-axis. 
For each value of X he computed the corresponding y from the 
equation, thus getting the coordinates x and y. The use of two 
axes obviously is not a necessity but a convenience. In our 
terminology, he used the equivalents of both rectangular and 
oblique axes. But in one important particular his procedure was 
needlessly restricted. He considered equations only in the first 
quadrant, as it was thence that he translated the geometry into 
algebra. This consistent but unnecessary limitation led to inex- 
plicable anomalies in the translation back from algebra to 
geometry. As analytic geometry evolved and negative numbers 
were fearlessly used, the restriction was removed. By 1748, when 
Euler codified and extended the work of his predecessors, both 
plane and solid analytic geometry were practically perfected, 
except for the introduction of homogeneous coordinates in 1827. 


The new method was not fully appreciated by Descartes’ 
contemporaries, partly because he had deliberately adopted a 
rather crabbed style- When geometers did see what analytic 
geometry meant, it developed with great rapidity. But it was 
only with the invention of the calculus that analytic geometry 
came into its own. As early as 1704 Newton 7 was able to classify 
all plane cubic curves into seventy-eight species, of which he 
exhibited all but six. This comparatively early work in the 
geometry of higher plane curves is especially remarkable for its 
discussion of the nature of the curves at infinity, and for New- 
ton’s assertion, which he did not elucidate, that all species are 
obtainable as projections (‘shadows’) of the curves y- = ax z 
+ bx- 4- cx 4- d. When we reflect that, only sixty-seven years 
before Newton published this work, geometers had been labori- 
ously anatomizing those other ‘shadows, 5 the conic sections, by 
the synthetic method of Apollonius and had not even imagined 
Newton’s cubics, we begin to appreciate the magnitude of the 
revolution Descartes precipitated in geometry. 

It is evident from Descartes’ explanation of his method that 
he had an intuitive grasp of the elusive concepts ‘variable’ and 
‘function,’ both of which are basic in analysis. Moreover, he 
intuited continuous variation. Vieta before him had used letters 
to denote arbitrary constant numbers; Descartes knew that the 
letters in his equations represented variables, and he clearly 
recognized the distinction between variables and arbitrary 
constants, although he defined neither formally. The significance 
of this advance for the calculus that was to follow only sixteen 
years after his death is plain. 

Descartes’ progress in generality is illustrated by two of his 
minor but geometrically important observations. He classified 
algebraic curves according to their degrees, and recognized 
that the points of intersection of two curves are given by solving 
their equations simultaneously. The last implies what actually 
is a major advance over all who had previously used coordinates: 
Descartes saw that an infinity of distinct curves can be referred 
to one system of coordinates. In this particular he was far ahead 
of Fermat, who, apparently, overlooked this crucial fact. Fermat 
may have taken it for granted, but nothing in his work shows 
unequivocally that he did. 

Still seeking generality, Descartes separated all curves into 
two classes, the “geometrical” and the “mechanical.” This is 
curious rather than illuminating. He defined a curve to be 


geometrical or mechanical according as (in our terminology) 
dy/dx is an algebraic or a transcendental function. Although this 
classification was abandoned long ago, it affords an interesting 
sidelight on the quality of Descartes’ mind. The current defini- 
tion of a transcendental curve as one which intersects some 
straight line in an infinity of points was given by Newton in 
his work on cubics. 

Descartes’ method for finding tangents and normals need not 
be described, as it was not a happy inspiration. It was quickly 
superseded by that of Fermat as amplified by Newton. Fermat’s 
method amounts to obtaining a tangent as the limiting position 
of a secant, precisely as is done in the calculus today. The his- 
torical significance of this conception is evident when it is recalled 
that the tangent at ( x , y ) is drawn by a simple Euclidean con- 
struction once its slope dy/dx is known. Fermat’s method of 
tangents is the basis of the claim that he anticipated Newton 
in the invention of the differential calculus. It was also the 
occasion of a protracted controversy with Descartes. 

We pass on to Fermat and his part in the invention of ana- 
lytic geometry. There is now no doubt that he preceded Des- 
cartes. But as his work of about 1629 was not communicated to 
others until 1636, and was published posthumously only in 1679, 
it could not possibly have influenced Descartes in his own inven- 
tion, and Fermat never hinted that it had. 

Fermat was one of those comparatively rare geniuses of the 
first rank, like Newton and Gauss, who find all their reward in 
scientific work itself and none in publicity. Under modern 
economic conditions, it is inexpedient for a scientist to hide his 
light under a bushel unless he wishes to starve to death in the 
dark, and few do. Of course it is impossible to say what effect the 
prospect of jobless starvation would have had on the more aloof 
scientists of the past. Some of them lived in economic security 
independently of any scientific work they might or might not do. 
Today men make the only livings they have at science or mathe- 
matics, and it seems like a misapplication of a warped yardstick 
to measure their professional ethics by those of a hypothetical 
past that may never have existed. For it has yet to be proved 
that a full mind can outargue an empty stomach. In Fermat s 
case, either lifelong security or excessive modesty made publica- 
tion of very minor importance to him, and as a result his superb 
talents were all but buried in his own generation. Descartes, not 
Fermat, was the geometer whom others followed. 


Pierre de Fermat (1601-1665, French, date of birth disputed) 
cultivated mathematics as a hobby. His profession, like Vieta’s, 
was the law. As a counselor of the local parliament at Toulouse, 
he lived a quiet, orderly life which left him ample leisure for 
his favorite study. An accomplished linguist and classicist as 
well as a first-rank mathematician, Fermat knew the master- 
pieces of Greek mathematics at first hand. 

Fermat did not discover his extraordinary powers in mathe- 
matics till he was about thirty, and even then he seems scarcely 
to have realized their magnitude. From his letters we get the 
impression that he regarded himself as a rather ingenious fellow, 
capable occasionally of doing a little better than Apollonius and 
Diophantus, but not very much after all in comparison with the 
ancient masters. Such sincere modesty would be engaging were it 
not exasperating: arithmeticians today would give a good deal 
for a glance at the methods Fermat must have devised but never 
published. In partial compensation for his indifference to publi- 
cation, Fermat was a voluminous correspondent. 

With the exception already noted concerning the use of one 
coordinate system for the representation of any number of 
curves, Fermat’s analytic geometry 8 appears to be as general as 
that of Descartes. It is also more complete and systematic. 9 By 
1629, according to Fermat’s own dating, which there is no reason 
to question, he had found the general equation of a straight 
line, the equation of a circle with center at the origin, and equa- 
tions of an ellipse, a parabola, and a rectangular hyperbola, the 
last referred to the asymptotes as axes. 

The year (1638) after Descartes had published his geometry, 
Fermat communicated to him the accepted method of finding 
tangents. This originated in Fermat’s investigation of maxima 
and minima, which he approached in substantially the same way 
as is done today in the calculus. What he did amounts to equat- 
ing the derivative f'(x ) of f(x) to zero to find the values of x 
which maximize or minimize f(x). Geometrically, this is equiva- 
lent to finding the abscissas of the points on the curve y = /(*) 
at which the tangent is parallel to the .r-axis. He did not proceed 
to higher derivatives or their geometrical equivalent to determine 
whether f(x) = 0 actually gives maxima or minima, as is 
necessary in a complete discussion. Nor did he isolate the calcula- 
tion of the derivative from its implicit occurrence in problems of 
maxima and minima. Descartes either did not grasp the supe- 
riority of Fermat’s method or was too chagrined to admit it, and 


liis side of the controversy over tangents became somewhat 

One positive gain has survived from this work on maxima 
and minima, Fermat’s principle of least time 10 in optics. 11 This 
was the first (1657, 1661) of the great variational principles of 
the physical sciences. 

As we shall pass on presently to Newton and his calculus, we 
may consider briefly here what the account given above of 
Fermat’s tangents implies. If accepted at its full value, it makes 
Fermat an inventor of the differential calculus. The greatest 
mathematician of the eighteenth century, Lagrange, did so 
accept it. But the verdict is not unanimous. 

The difference of opinion seems to hinge on Fermat’s implicit 
conception of initially ‘neighboring’ but ultimately coincident 
points on a curve. To maximize or minimize/^) Fermat replaced 
x by x -f- E, where E differs but little from zero. He then equated 
f(x) to f(x + E ), simplified the algebra, divided by E, and finally 
set E equal to zero. 12 

If this is legitimate differential calculus, then Fermat in- 
vented that calculus. If it is not, it seems no more illegitimate 
than its historical rivals. Thus Newton in his exposition of 1704, 
discussing the “fluxion” of x n , n an arbitrary rational number, 
used his binomial formula (1676) to expand ( x + o) n , and formed 
the difference (a: + o) n — x n . He then said, “Now let these 
augments [namely, ( x + o) n — x n and o] vanish, and their ulti- 
mate ratio will be 1 to nx n_1 .” This was his method of “prime 
and ultimate ratios.” In the Leibnizian notation used today, 
Newton thus finds dx n /dx = nx n ~ x . 

The conclusion seems to be that either nobody in the seven- 
teenth century invented the differential calculus, or Fermat was 
one of those who did. The matter is not settled by citing New- 
ton’s conception of a limit, because he did not develop a theory of 
limits in what he actually printed. But on this debatable differ- 
ence of opinion everyone must form his own opinion after 
understanding the evidence as he may. 

Before leaving the creators of analytic geometry, we may 
mention three further items from Descartes, although only the 
first is related to geometry. Descartes devised the notation 
x, xx, x z , x A , . . . for powers, and made the final break with the 
Greek tradition of admitting only first, second, and third powers 
(‘lengths,’ ‘areas,’ ‘volumes’) in geometry. After Descartes, 
geometers freely used powers higher than the third without a 


qualm, recognizing that representability as figures in Euclidean 
space for all of the terms in an equation is irrelevant to the 
geometrical interpretation of the analysis. 13 

The principle of undetermined coefficients was also stated by 
Descartes. Anything approaching what would now be admitted 
as a proof was about two centuries beyond the mathematics of 
his time. A second outstanding addition to algebra was the 
famous rule of signs given in even' text on the theory of equa- 
tions. This was the first universally applicable criterion for the 
nature of the roots of an algebraic equation. Even if it does not 
always yield any useful information, it admirably illustrates 
Descartes’ flair for generality which made him the mathe- 
matician he was. 

Newton, Leibniz, and the calculus 

In the history of Newton’s calculus, the temptation to read 
‘anticipation’ into the works of his contemporaries and immedi- 
ate predecessors is perhaps stronger than in that of any other 
major advance in mathematics. Knowing what wc now do of 
the calculus and its implications in geometry and elementary 
kinematics, wc can look back on many isolated discoveries in 
those domains and see in them what we now recognize as steps 
toward differentiation. But the discoverers, sometimes to our 
amazement, completely missed what wc now perceive so plainly. 
They failed in each instance to take the last gigantic stride that 
now seems to us but a short step; and to credit them with strides 
they might have taken but did not is sheer sentimental 

As a relevant exercise in distinguishing between mathe- 
matical insight and facile prophecy after the fact, students of 
the calculus may wish to test their critical powers on the history' 
of the “differential triangle” of Isaac Barrow (1630-1677, 
English). This was somewhat in the manner of Fermat. Ignoring 
this, we shall adhere to the generally accepted tradition and 
proceed on the hypothesis that Newton in his calculus did some- 
thing new. 

Isaac Newton (1642-1727, English), the posthumous son of 
a yeoman farmer, was bom near Grantham, Lincolnshire, and 
passed his boyhood there. As a boy he was only passively inter- 
ested in his schooiwork until he suddenly woke up at the age of 
adolescence. Earlier, he had shown unmistakable promise of 
experimental genius in the mechanical toys he invented and 


made to amuse himself and his young friends. It is interesting 
that both Newton and Descartes were delicate in childhood, and 
therefore had time to think and develop their own personalities 
while rougher boys were reducing one another to a very common 
denominator. Both matured into sturdy men, Descartes through 
military training, Newton by the inherited toughness of his 
farmer forebears. 

After a desultory attempt to learn farming, Newton was sent 
to Trinity College, Cambridge, in 1661 (age nineteen). His 
undergraduate career, from all that is definitely known of it, was 
not particularly distinguished. Before going to Cambridge he 
had skimmed Euclid’s Elements , and is said to have dubbed it 
“a trivial book.” When, later, he understood Euclid’s purpose, 
he revised his hasty judgment. In his own work he refers to 
Euclid with evident respect. His baffling encounters with “very 
little quantities” made him appreciate at least the tenth book 
of the Elements. It should encourage intelligent beginners to 
know that Newton found analytic geometry difficult at a first 

Perhaps fortunately for mathematics, Newton’s studies were 
interrupted in 1665-6 by the Great Plague, when the university 
closed. Newton returned home, but not to farm. Before he was 
twenty-four years of age he had imagined the fundamental ideas 
of his fluxions (calculus) and his law of universal gravitation. 

On returning in 1667 to Cambridge, Newton was elected a 
fellow of Trinity, and in 1669 succeeded Barrow, who resigned in 
his favor as Lucasian professor of mathematics. His first work to 
become known beyond the narrow circle of his intimate friends 
was in optics, beginning with his lectures of 1669. 

As we are interested here mainly in Newton’s calculus, we 
shall merely summarize the material circumstances of his career, 
full accounts of which are readily accessible. These also describe 
his epochal work in optics, which will not be discussed here as 
it belongs rather to physics than to mathematics. 

In 1672 (age thirty), Newton was elected to the Royal 
Society, and from 1703 till his death was its president. His 
Principia, universally estimated by competent judges to be the 
greatest contribution to science ever made by one man, was 
composed in 1684-6 at the instigation of the astronomer E. Hal- 
ley (1656-1742), at whose expense it was printed in 1687. 

In 1689, and again in HOT Newton was elected to represent 
Cambridge University in Parliament. He had no taste for de- 


bating, but he took his duties seriously, and showed a fine cour- 
age in championing the University’s rights against the dictatorial 
meddling of King James the Second. At the age of fifty (1692) he 
suffered a severe illness and lost interest in scientific work, 
although he retained his unsurpassed intellectual powers to the 
end of his life. 

Partly by his own desire, partly at the insistence of friends 
who wished to sec him honored, Newton entered public life when 
he tired of science, and in 1696 was made warden of the mint. 
Having successfully directed the reform of the coinage, he was 
promoted to the mastership in 1699. In 1705 he was knighted by 
Queen Anne, and in 1727 he died. He is buried in Westminster 

Newton’s excessive reluctance to publish his scientific work 
reflects certain aspects of his character. Although by no means 
a shy or timid man, Newton had a strong distaste for anything 
bordering on controversy. An unintelligent dispute over his work 
in optics at the beginning of his career taught him that scientific 
men arc not always so objective as they might be, even in 
science, and he retired within himself in astonished disgust. 

Nor was his notorious indifference to the survival of his 
scientific work affectation. But for the adroit coaxing and goad- 
ing of Halley, the Principia would probably never have been 
written. Newton himself esteemed the theological writings to 
which lie devoted the leisure of his later years far more highly 
than his science and mathematics. Again, in his work on light, 
Newton had proved himself one of the most acute experi- 
mentalists in the history of science; so it was but natural that 
he should spend much time and a considerable amount of 
money on what we should call alchemy, but what in his day 
was orthodox chemistry. 

It was the ironic misfortune of this hater of profitless dis- 
putes to be embroiled in the most disastrous mathematical con- 
troversy in history, when some of his busy friends inveigled him 
into insinuating that Leibniz had plagiarized his own form of the 
calculus. We shall not discuss this, but merely state that the 
almost universal opinion now is that Leibniz invented his calculus 
later than Newton and independently. It must be gratifying to 
Englishmen to recall that it was another English mathematician, 
that born nonconformist Augustus De Morgan (1S06-187 j), 
who first undertook a judicial examination of the dispute and 
obtained some measure of justice for Leibniz. 


ability, and his lack of a feeling for tangible things occasionally 
betrayed him in science. Like Descartes, M who also went astray 
in science, Leibniz is probably most widely known today for his 
philosophy; but to a modern scientific mind his monads are as 
fantastically absurd 15 as Plato’s eternal ideas. He thought 
incessantly. His unresting curiosity was attracted by everything 
and distracted by nothing. Perhaps the world is fortunate that 
much of his intelligence was dissipated in one way or other in the 
pursuit of money and fugitive honors. 

As the reward for a revolutionary essay on the teaching of 
the law, Leibniz at the age of twenty-one was engaged by the 
Elector of Mainz as general agent and legal adviser. Most of his 
time thereafter was spent in travel on diplomatic missions for 
the Elector until the latter’s death in 1673. Leibniz then became 
librarian, historian, and political factotum for the Brunswick 
family at Hanover. 

During visits to France and England on political or diplo- 
matic missions, Leibniz met the leading French and English men 
of science, and in exchange for some of their ideas disclosed his 
own. One such trade was to prove profoundly significant in the 
development of the calculus. If we seek the origin of modern 
work in the foundations not only of analysis but of all mathe- 
matics, we need look no farther than the following incident. 

Until lie met the great Dutch physicist and mathematician 
Christian Huygens (1629-1695) in Paris in 1672, Leibniz had but 
little if any competence in what was then modern mathematics. 
Such firsthand mathematical knowledge as lie had was mostly 
Greek. Huygens enlightened him and undertook his mathe- 
matical education. Leibniz proved himself an exceedingly apt 
pupil. The two became good friends, corresponding till the 
death of Huygens in 1695. Leibniz begged Huygens for criticism 
of his projects and, naturally, got it. 

It is only a speculation, but from Leibniz’ ambitious char- 
acter and his philosophic propensity for solving the universe, it is 
conceivable that his daring project for a universal symbolic 
reasoning was fostered by a determination to beat Descartes 
at his own game. The philosophic Frenchman had reduced all 
geometry to a universal method; the more philosophic German 
would similarly reduce ali reasoning of whatever kind to a 
universal “characteristic” or, as would be said today, a symbolic 
mathematical science. Leibniz in 1679-80 confided his project 
to Huygens. sc The physicist was not impressed. 


By fatal mischance, Leibniz chose a trivial and singularly 
uninteresting geometrical problem to illustrate what he in- 
tended , 17 with the result that Huygens misunderstood the entire 
matter. He became somewhat polemical. This failure to see 
what Leibniz meant is the more remarkable as Huygens himself 
had a scientific vision which saw forests in spite of their innum- 
erable trees. Possibly he was antagonized by Leibniz’ boastful 
attitude. In his misunderstanding of what the ambitious phil- 
osopher-mathematician was trying to do, Huygens for once 
descended to the pedantries of captious criticism. 

At first glance it may seem that Leibniz’ attempts toward 
symbolic logic are irrelevant in the development of the calculus. 
Nothing could be farther from the fact. We shall see presently 
that Newton in his early encounters with continuity lost himself 
in the racecourse of Zeno, of whose paradoxes he perhaps had 
never heard. Subtly disguised but yet the same, these hoary 
difficulties have perplexed every mathematician, from Newton 
in the seventeenth century to Weierstrass in the nineteenth, 
who has sought not merely to obtain useful or interesting results 
by routine differentiations and integrations, but to understand 
the calculus itself. The calculus was difficult to Newton and 
Weierstrass; it is easy only to those who understand it too 

The modern attack on the fundamental problems of con- 
tinuity has revealed the nature of the difficulties which baffled 
Newton, Leibniz, and the more thoughtful of their successors. 
It seems safe to say that without the mathematical logic which 
Leibniz advocated, and which he started to create, the critical 
work of the twentieth century on the foundations of analysis, and 
indeed of all mathematics, would have been humanly impossible. 

Leibniz imagined the project of a ‘calculus’ of deductive 
reasoning; and if his own steps toward it weje but short and 
hesitating, nevertheless it was his bold conception which encour- 
aged others to proceed. It seems rather late in* the day, therefore, 
to persist in seeing Leibniz the mathematician merely as a major 
satellite of Newton. / 

Historical tradition reiterates to weariless the undisputed 
fact that the ^-notation of Leibniz is vastly) superior to the dots 
of Newton. But if, with Gauss , 18 we believe {that in mathematics 
notions are more important than notations, ( we must place the 
emphasis elsewhere. The greatest work of Leibniz, from the 
standpoint of modern mathematics, is not hi's improvement of 


the differential and integral calculus, great though that was, but 
his calculus of reasoning. He shines by his own light. 

Little need be said here of Leibniz’ career as a diplomat. 
To him is attributed that epitome of unstable equilibrium which 
later jugglers of destiny were to worship as the Balance of Power. 
In his diplomacy, Leibniz was neither more nor less unscrupulous 
than any of his famous successors in that dubious art. He was 
merely less incompetent than the majority. Unlike some of them, 
Leibniz did not succumb beneath a weight of honors heaped on 
him by grateful princes, but died neglected and forgotten by 
those whose petty fortunes he had made. When his employer 
departed from Hanover to become King George the First of 
England, Leibniz was discarded in the library to continue his 
history of the Brunswick family, surely a fitting occupation for 
one of the supreme intellects of all time. Only his secretary 
followed him to his grave. 

Such were the two mortals who finally created the calculus. 

Newton’s version of the calculus 

Newton’s first calculus, of 1665-6, seems to have been 
abstracted from intuitive ideas of motion. A curve was imagined 
as traced by the motion of a ‘flowing’ point. The ‘infinitely 
short’ path traced by the point in an ‘infinitely short’ time was 
called the "momentum” and this momentum divided by the 
infinitely short time was the "fluxion.” If the “flowing quan- 
tity” is .v, its fluxion is denoted by A. In our terminology, if .v 
is the function /(t) of the time t, x is dx/dt, the velocity at time 
!. Similarly, the fluxion of .v is x, our d-x/dt-] A is d 3 x/dl 3 , and 
so on. 15 

Newton regarded our dxfdt as the actual ratio of two “in- 
finitely small quantities” in this first calculus. He had no 
approach to a limit that would be recognized today. The follow- 
ing extract from the Principia (1687) will indicate that Newton 
himself was dissatisfied with his own refinement of the method 
of fluxions. 

It is objected that thercis no ultimate ratioof evanescent quantities because 
the proportion (ratio] before the quantities have vanished is not ultimate; and, 
when they have vanished, is none. But by the same argument, it might as well 
be maintained, that there is no ultimate velocity of a body arriving at a certain 
place, when its motion is ended: because the velocity, before the body arrives 
at the place, is not its ultimate velocity; when it has arrived, is none. But the 
answer is easy . , . There is a limit, which the velocity at the end of the 
motion may attain, but cannot exceed. 


This is what Zeno and the tortoise knew, and what neither of 
them succeeded in clarifying. It is no disparagement of Newton, 
to observe that the foregoing extract might have been written 
by Aristotle; indeed it bears a singular resemblance to Aristotle’s 
discussion” of the infinite, the continuous, motion, and Zeno’s 
paradoxes. A further observation of Newton’s recalls Eudoxus: 

“It may also be argued, that if the ultimate ratios of evanes- 
cent quantities are given, their ultimate magnitude will also be 
given; and so all quantities will consist of indivisibles, which is 
contrary to what Euclid has demonstrated concerning incom- 
mensurables, in the tenth book of his Elements .” Again compare 
with Aristotle. 

From the last it is clear that Newton’s understanding of 
Euclid was sharper than Cavalieri's. It also suggests that the 
difficulties of intelligent beginners with limits and continuity 
are not mere willful perversity. In his third attempt (1701), 
Newton returns to the attack on continuity, and transfers the 
central difficult)' to an unanalyzed “continued motion”: 

“I consider mathematical quantities in this place not as con- 
sisting of very small parts; but as described by a continued 
motion. Lines are described, and thereby generated not by 
the apposition of parts, but by the continued motion of points 


* * • 

It was considerations of this kind, among others, that drove 
the analysts of the nineteenth century to desperation and im- 
pelled them to attempt a meaningful foundation for the calculus. 
In spite of his resolute abandonment of “very small parts,” 
Newton never quite circumvented the “very' small quantity’’ 
which persistently annoyed him. In the Principia (Bk. I, See. I, 
Lemma I), he started toward a theory of limits and continuity: 

“Quantities, and the ratios of quantities, which, in any 
finite time, tend continuously to equality; and before the end 
of that time, approach nearer to each other than by any given 
difference, become ultimately equal. r 

Leibniz’ version 

Leibniz for his part favored a species of differential, as that 
highly elusive concept is frequently misunderstood by practical 
engineers today. Thus, to find the differential o: xy, he sub- 
tracted xy from (x -j- dx') (y -f dy) and rejected dx dy because 
he considered it negligibly small in comparison with x dy ana 


y dx — nil without sound justification. This gave him the correct 
result, d{xy) — x dy -f- y dx. 

On safer ground, lie introduced the current notation for 
derivatives and the integral sign, J, an elongated s from sutnma 
(sum). Both Leibniz and Newton were familiar with the funda- 
mental theorem of the calculus connecting integrals as sums 
with integrals as anti-derivatives. They also established the 
elementary formulas of the calculus. It is of interest that 
the correct result for the derivative of a product eluded Leibniz 
on his first attempt. 

Rigor; anticipations 

It is generally agreed that reasonably sound but not neces- 
sarily final ideas of limits, continuity, differentiation, and 
integration came only in the nineteenth and twentieth centuries, 
beginning with Cauchy in 1821 -3. This raises an extremely 
interesting question: how did the master analysts of the eight- 
eenth century — the Bernoullis, Euler, Lagrange, Laplace — 
contrive to get consistently right results in by far the greater 
part of their work in both pure and applied mathematics? 
What these great mathematicians mistook for valid reasoning 
at the very beginning of the calculus is now universally regarded 
as unsound. 

No short answer is possible; but history shows that frequently 
the essential, usable part of a mathematical doctrine is grasped 
intuitively long before any rational basis is provided for the 
doctrine itself. The creative mathematicians between Newton 
and Cauchy obtained mostly correct results — according to 
present standards — because, in spite of their ineffectual attempts 
to be logically rigorous, they had instinctively apprehended the 
self-consistent part of their mathematics. 

Just as no short answer can dispose of our predecessors’ good 
fortune, so none can dispose of ours. Like them, we consistently 
get meaningful results, although we realize that there is much 
obscurity in the foundations of our own analysis. It is now 
generally admitted that neither Cauchy nor his more rigorous 
successor Weicrstrass said the last word, and we may con- 
fidently expect that it will not be uttered in our generation. 

Whatever else may be said of Newton’s calculus, it is still 
true that he endowed mathematics and the exact sciences with 
their most effective method of exploration and discover}'. 
Linked to his own law of universal gravitation, the calculus in 


less than a century gave a more comprehensive understanding 
of the solar system than had accrued from thousands of years 
of prc-dynamical astronomy. And when differential equations, 
Newton’s method of inverse tangents, were applied to the 
physical sciences, a new and unsuspected universe was revealed. 
The experimental method of Galileo combined with the calculus 
of Newton and Leibniz generated modern physical science and 
its applications to technology. 

To conclude this account of the emergence of the calculus, wc 
shall compensate for the deliberate neglect of pseudo anticipa- 
tions by citing a real one in a subject which is basic for the 
modern attack on the foundations of analysis. 

Galileo observed as early as 1638 that there are precisely 
as many squares 1, 4, 9, 16, 25, ... as there are positive in- 
tegers altogether. This is evident from the sequences 

1, 2, 3, 4, 5, 6,. . . , . . . 

I 2 , 2 2 , 3 2 , 4 2 , 5 2 , 6 2 , . . . , 71 2 , . . . 

He thus recognized the fundamental distinction between finite 
and infinite classes that became current in the late nineteenth 
century. An infinite class is one in which there is a one-one 
correspondence between the ■whole class and a subclass of the 
whole. Or, what is equivalent, there are as many things in some 

part of an infinite class as there are in the whole class. The like 

is not true of finite classes. 

A class whose elements can be put in one-one correspondence 
with the integers 1, 2, 3, . . . is said to be denumerable. All the 
points in any line segment, finite or infinite in length, form a 
non-denumerable set. A basic course in the calculus (usually 
the second) starts from the theory of point sets. The distinction 
between denumerable and non-denumerable classes was not 
stated by Galileo; it was observed about 1840 by Bolzano and in 
1878 by Cantor. But Galileo’s recognition of the cardinal prop- 
erty of all infinite classes makes him one of the genuine anticipa- 
tors in the history of the calculus. The other was Archimedes. 

Emergence of the mathematical theory of probability 

Games of chance are probably as old as the human desire 
to get something for nothing; but their mathematical implica- 
tions rvere appreciated only after Fermat and Pascal in I6a4 
reduced chance to law. Galileo had given a correct solution oi a 
gaming problem by laborious tabulation of possible cases, but 


did not proceed to general principles. The “problem of points” 
which inspired the originators of the mathematical theory of 
probability was well known to Cardan who, among his other 
accomplishments, was a reckless gambler. He, however, did 
nothing of importance toward a science of chance; and it is 
customary without any quibbling to regard Pascal and Fermat 
as the founders of mathematical probability. 

In the epoch-making problem, the first of two players who 
scores n points wins. If the game is abandoned when one has 
made a points and the other b, in what ratio shall the stakes be 
divided between them? This reduces to calculating the prob- 
ability each has of winning when the game is stopped. It is 
assumed that the players have equal chances of making a point. 

The problem was proposed to Pascal by a highly intelligent 
gentleman addicted to gaming, Antoine Gombaud, Chevalier 
dc Mere, and Pascal communicated it to Fermat. Both solved 
it correctly, but by different reasoning. In some of his work, 
Pascal made a slip, which Fermat corrected. Thus originated the 
mathematics of chance which today is basic in all statistical 
analysis from stock-market trends and insurance to intelligence 
tests and biometrics. 

As modern physics has become more certainly uncertain, the 
mathematics of probabilities has steadily increased in scientific 
importance. Newtonian mechanics is applicable to a completely 
determinate science in which differential equations imply the 
future history of a mechanistically determined universe. For the 
scientific interpretation of laboratory experiments, particularly 
in atomic physics, the strictly mechanical method of Newton, 
Lagrange, Laplace, and their successors, originating in Galilean 
mechanics and dynamical astronomy, is no longer adequate and 
is being increasingly supplemented by the mathematics of 
statistics and probability. The necessary mathematics all 
developed from the fundamental principles of mathematical 
probability laid down by Fermat and Pascal in about three 
months by a painstaking application of uncommon sense. 21 

Later applications of analysis to the theory of probability, 
chiefly to obtain usable approximations to the very large num- 
bers occurring in even simple combinatorial problems, have 
made the modern theory highly technical. But with the excep- 
tion of epistemological difficulties concerning the meaning of 
probability, the basic principles remain those of 1654 as stated 
in intermediate texts on algebra. In this connection it should be 


mentioned that the versatile Huygens got wind of what Pascal 
and Fermat were doing, and in 1654 published one of the earliest 
treatises on probability. 22 The concept of mathematical expecta- 
tion was his. 

The relative permanence of the mathematical foundations of 
probability as laid down in the seventeenth century is character- 
istic of the mathematics of the discrete, in which generally there 
has been less need for revision than in analysis. 

The origin of modern arithmetic 

We shall understand ‘arithmetic 5 in the sense of the Greek 
arithmetica. Equivalents are ‘the higher arithmetic 5 and ‘the 
theory of numbers, 5 also, unfortunately, the hybrid ‘number 
theory 5 with its Aryan adjective and adverb, ‘number theoretic 5 
and ‘number theoretically. 5 Gauss, the foremost exponent of the 
classical theory after Fermat, preferred the simpler ‘arithmetic 5 
or, at longest, ‘higher arithmetic. 5 

Modern arithmetic began with Fermat, roughly in the 
period 1630-65. Significant as was Fermat’s work in other 
departments of mathematics, he is usually considered to have 
made his greatest and most personal contribution in arithmetic. 

This extensive division of mathematics differs from others in 
its lack of general methods. Even comprehensive theorems 
appear to be more difficult to devise than, say, in algebra or 
analysis. Thus, in algebra there is a complete theory of the 
solution of algebraic equations in one unknown; in fact there 
are two complete theories. In arithmetic the simplest corre- 
sponding problem is the solution.! in integers of equations in two 
unknowns with integer coefficients, and for this there is nothing 
approaching a complete theory,. Such progress as has been made 
since Fermat will be noted in £-1 later chapter. 

Many of Fermat’s discoveries were either recorded as 
marginal notes in his books 1 ! (the arithmetic in his copy of 
Bachet’s Diophantus) or were communicated, usually without 
proof, to correspondents. Somle of his theorems were proposed by 
him as challenge problems to the English mathematicians. For 
example, he demanded a proof that the only positive integer 
solution of x 1 + 2 = y 3 is x — 5, y = 3. 

It will suffice to state those two of Fermat’s discoveries 
which appear to have had the profoundest influence on arith- 
metic and algebra since his time, ‘'and the one general metho m 
arithmetic due to him. 


Fermat stated that if ?! is a positive integer not divisible by 
the positive prime p, then n p_1 — 1 is divisible by p. The Chinese 
“seem to have known as early as 500 B.c.” s * the special case 
it *= 2. Any student of the theory of algebraic equations, or of 
modern algebra, or of arithmetic, will recall the frequent appear- 
ance of this fundamental theorem. The first published proof 
was Euler’s in 1738, discovered in 1732; Leibniz had obtained 
a proof before 1683 but did not publish it. The rule of priority in 
mathematics is first publication. 

The second famous assertion of Fermat, his celebrated ‘Last 
Theorem,* states that xF -f- y n — z n , xyz A- 0, n > 2, is im- 
possible in integers x, y, z, n. He claimed (1637) to have dis- 
covered a marvelous proof; and whether or not he had, no proof 
has yet (1945) been found. There seems to be but little point 
now in proving the theorem for special ?i’s, enough in that direc- 
tion being known to make it fairly plausible that the theorem is 
true. But, to take out insurance against a possible disproof 
tomorrow, it must be emphasized that arithmetic is the last 
place in mathematics where unsubstantiated guessing is either 
ethical or profitable. Numerical evidence counts for very little; 2 * 
the only luxury a reputable arithmetician allows himself is 

It is generally agreed that the famous ‘Last Theorem,* true 
or false, is of but slight interest today. But its importance in the 
development of arithmetic and modern algebra ha3 been very 
great. This will be discussed in the proper place. 

Fermat’s general method, that of “infinite descent,” is pro- 
foundly ingenious, but has the disadvantage that it is often 
extremely difficult to apply. In the particular theorem for which 
Fermat invented the method, it is required to prove that every 
positive prime of the form 4« -f- 1 is a sum of two integer squares. 
From the assumption that the theorem is false for some such 
prime p, Fermat deduced that it is also false for a smaller prime 
of the same kind. Descending thus he proved, on the assumption 
of falsity, that 5 is not the sum of two squares. But 5 — 1 2 -f- 2 2 ; 
hence the theorem. 

The outstanding desideratum in arithmetic is the invention 
of general methods applicable to nontrivial types of problems. 
Further, “the arithmetical solution of a problem should consist 
in prescribing a finite number of purely arithmetical operations 
(exempt from all tentative processes), by which all the numbers 
satisfying the conditions of the problem, and those only arc 


obtained.” 1 * Nobody after Euclid and before Lagrange in the 
eighteenth century even distantly approached this ideal. 

Emergence of synthetic projective geometry 

The sudden rise of synthetic projective geometry in the 
seventeenth century appears now as a belated resurrection of 
the Greek spirit. As already noted, Pappus in the fourth century 
a.d. anticipated a cardinal property of cross ratios; and even 
earlier Menelaus (first century a.d.) may have proved a theorem 
which can now be interpreted similarly. But it was only with 
G. Desargues’ (1593-1662, French engineer and architect) 
eccentric Brouillon project (abbreviated title) of 1639 that 
synthetic projective geometry' was developed as a new and 
independent division of geometry. 

Doubtless the great advance in perspective drawing by the 
artists of the Renaissance made inevitable the emergence of a 
geometrical theory including perspective as a special case; and 
Desargues the architect was doubtless influenced by what in 
his day was surrealism. In any event, he composed more like an 
artist than a geometer, inventing the most outrageous technical 
jargon in mathematics for the enlightenment of himself and the 
mystification of his disciples. Fortunately, Dcsargucsian has 
long been a dead language. In current terminology, ‘projective’ 
means invariance under the group G of all general linear homo- 
geneous transformations in the space (of 1, 2, 3, . . . dimen- 
sions) concerned, but not under all the transformations of any 
group containing G as a subgroup. 

After his own fashion, Desargues discussed cross ratio; 
poles and polars; Kepler’s principle (1604) of continuity, in 
which a straight line is closed at infinity and parallels meet 
there; involutions; asymptotes as tangents at infinity; his 
famous theorem on triangles in perspective; and some of the 
projective properties of quadrilaterals inscribed in conics. 
Descartes greatly admired Desargues’ invention, but happily 
for the future of geometry did not hesitate on that account to 
advocate his own. 

Desargues’ most enthusiastic convert was the same Pascal 
who participated in the creation of the mathematical theory' of 
probability'. B. Pascal (1623-1662) was therefore a very con- 
siderable mathematician, even if, like Descartes and Leibniz, 
he is popularly remembered for other things. His magnitude as a 
religionist has overshadowed his accomplishments as a mathe- 


matician and physicist, and for one who has ever heard of 
Pascal’s Essay pour Ies eoniques there must be a million who have 
read at least a page of his Pensees. If anything, Pascal was more 
genuinely precocious than Leibniz. As a boy he was no mere 
sponge absorbing the learning of others, but a creative mathe- 
matician. At twelve he rediscovered and proved for himself 
several of the simpler theorems of elementary geometry. Four 
years later he had composed the famous essay on conics, in 
which he developed the consequences of his hexagramma mysti- 
cum — pairs of opposite sides of a hexagon inscribed in a conic 
intersect in collinear points. Combining his mathematics and his 
talent for physics, Pascal at nineteen (16-12) invented an adding 
machine, the ancestor of all those in use today. This was greatly 
improved about thirty years later by Leibniz, whose machine 
did both addition and multiplication. 

Pascal made grateful acknowledgment to Desargues for his 
skill in projective geometry. Perhaps in all of his mathematics 
Pascal was the brilliant commentator rather than the bold 
originator. Organically and spiritually ill for most of his thirty- 
nine years, he was unable, apparently, to concentrate his powers 
on the creation of a comprehensive method in anything, and his 
brilliance was dispersed and dissipated in the piecemeal illumina- 
tion of other men’s ideas. Much of his mentality was absorbed in 
the religious controversies of his time and in hopeless attempts 
to reconcile his own internal conflicts. Beyond his ‘mystic 
hexagram’ and his share in probability, it cannot be said that 
Pascal’s contributions left more than a transient shadow on the 
surface of mathematics. The main stream flowed far deeper than 
he ever dreamed. 

Before quitting the field for about a century and a quarter, 
synthetic projective geometry' fought a terrific pitched battle to 
survive against its analytic antagonist, in the impressive Sec- 
liones co niece (1685) of P. de ia Hire (1610-1718, French). 
La Hire proved over three hundred projective theorems syn- 
thetically, and in an astounding appendix showed that all the 
theorems of Apollonius on conics arc obtainable by the method 
of projection. But even these spectacular gymnastics failed to 
convince geometers that the synthetic method is as supple as the 
analytic. The conics no doubt were idealized from the Archetypal 
Circle by Plato’s Geometer with the Eternal Idea of projective 
geometry’ at the back of his mind; but then, not all plane curves 
arc conics. Nor were they even when La Hire fought his desperate 

rearuunrd action npnin't. De* carte*’ analytic*. Syrtlwrir - r -„ 
jective geometry iap £ cd into ttmr^rr.rv cblivix;. ;• r. d - 
treat he* of Dc^nrpuc* and La Hire became emir :tor j ’ ra-hh-. 

The other incidental advance of the seventeenth t+r.uitv 
that likcwbe war forgotten for a re a* ‘in. the uni vernal char; 

Ltic of Leibniz, wiil be noted in a later chapter. We pr.", , t ) 
the oritrin of the applied mathematics which wa*- to dominate t; - 
work of Newton's most prominent successors for a cctuurv r:V* 
his death. 

Or:g:r. of rr.odrn 

s* * +• J-V 
**/ a 

The indebtedness of science and technology to pare rr.athr- 
m a tics was noted in the Prospectus. We shall now impret the 
other ride of the ledger at somewhat greater length than rruy 
seem necessary to those already conversant with the fact*. We 
do this because hypersensitive mathematician'* arc nuwttr.t: 
inclined to exalt unduly the freedom and purely unarT.rVAe 
charactcr of their creations, and to dwell exclusively on v r 
admitted indebtedness of science to mathematic:. Tim hb- 
lorical balance sheet indicates, as will be seen in greater sb-tril 
later, that science and modem mathematics are so clwriy 
nfhliatcd that neither owes the other anything, each borrow::,,- 
freely from the other and repaying its debt 1 - a hundredfold. 

Intermediate between pure and applied mathematic** Im- 
provements in numerical calculation arc of more imp 5 * 
the applications than for mathematics itself. \s varkhnw, Lr 
example, accelerated the practical development* t.f ro’.ror. my 
but were not a necessity even to that most efficient servant >•: 
civilization. The pertinacity of a Kepler cannot be thwart' ; :y 
any amount of manual computation; and to claim that lv> 
rithms made modern astronomy or any other p c:<-nce p.‘ w c 
is to forget that human zeal — or ob'tinacy — in purs';:: o'. a 
fixed idea can withstand any finite punishment. But a' 
rithms undoubtedly hastened the sciences of the cmht 
nineteenth centuries on their way to whatever is :■ 
ultimate contribution to civilization, or to its dftruct:' 
must be included in any account of the origin of 
mathematics. The seventeenth-century invcntvn o* ;* v 
may therefore be properly assigned to applied mati.crr.a 

Modern applied mathematic* oririnated in Newt: ‘ 
of universal gravitation developed in his P'inr.r :e. 
before Newton wa* purclv descriptive. Tr.c 



planets were described with increasing accuracy, and from the 
Babylonians to Ptolemy were fitted into geometrical frameworks 
of ever greater complexity. Copernicus simplified the geometry. 
But there was no physical hypothesis abstracted and con- 
solidated in postulates from which the geometry could be 
deduced. Before such postulates could be stated profitably, 
accurate observations to determine the facts were necessary. 
These were provided in abundance by Tycho Brahe (15*16-1601, 
Danish), whose industrious assistant for a short time, Johann 
Kepler, (1571-1630, German), subsumed the observations under 
the three laws of motion known by his name. The first two were 
published in 1609, the third in 1619: the orbit of a planet is an 
ellipse with the sun at one focus; the areas swept out in equal 
times by the line joining the sun to a planet are equal; the 
squares of the periodic times of the planets are proportional to 
the cubes of their mean distances from the sun. 

Kepler’s laws were the climax of thousands of years of an 
empirical geometry of the heavens. They were discovered as the 
result of about twenty-two years of incessant calculation, with- 
out logarithms, one promising guess after another being ruth- 
lessly discarded as it failed to meet the exacting demands of 
observational accuracy. Only Kepler’s Pythagorean faith in a 
discoverable mathematical harmony in nature sustained him. 
The story of his persistence in spite of persecution and domestic 
tragedies that would have broken an ordinary man is one of the 
most heroic in science. 

The contemporaneous invention of logarithms was to 
reduce all such inhuman labor as Kepler’s to more manageable 
proportions. The history of logarithms is another epic of per- 
severance second only to Kepler’s. Baron Napier of Merchistoun 
(1550-1617, Scotch), in the leisure remaining from his duties 
as a landlord and his unavailing labors to prove that the reigning 
pope was Antichrist, invented logarithms. 

When it is remembered that Napier died before Descartes 
introduced the notation nil, n, s ... for powers, we cease to 
wonder that it took him all of twenty years to reason out the 
existence and properties of logarithms. 

The fundamental idea of the correspondence bctv.-ccn two series of numbers, 
one in arithmetic, the other in geometric progression, . . . was explained by 
Napier through the conception of two points marine on separate straight lines, 
the one with uniform, the other with accelerated velocity. If the reader, with 
all his acquired modern knowledge, will attempt to obtain for hinswlf in this 


way a demonstration of the fundamental rules of logarithmic calculation he 
will rise from the exercise with an adequate conception of the penetrating 
genius of the inventor of logarithms. (G. Chrystal.) 

Add to this that Napier’s logarithm of n would be our 
10 7 loge (10 7 ft" 1 ), where e is the base of the natural system. 

After the invention of the calculus, investigation of the 
logarithmic function, of greater significance in mathematics 
than the logarithms computed by its use, followed as a matter 
of course from the simple differential equation dy = y dx. 

Napier gave Tycho a forecast of his invention in 1594, and 
in 1614 published his Descriptio. In 1624 a usable table by 
H. Briggs (1561-1631, English) was published, as also was one 
by Kepler. Other tables quickly appeared, and by 1630 loga- 
rithms were in the equipment of every computing astronomer. 

For those interested in squabbles over priority, it may be 
recalled that logarithms are one of the most disorderly battle- 
grounds in mathematical history. It will suffice here to state the 
outcome of the fray as adjudicated in 1914. Napier’s priority in 
publication is undisputed; J. Biirgi (1552-1632, Prague) inde- 
pendently invented logarithms and constructed a table between 
1603 and 1611, while “Napier worked on logarithms probably 
as early as 1594 . . . ; therefore, Napier began working on 
logarithms probably much earlier than Biirgi.” 26 The only facts 
concerning logarithms of any importance for the development 
of mathematics are those stated in the concluding sentence of 
the preceding paragraph. 

Disputes like this and the other over the calculus have made 
more than one man of science envy his successors of ten thousand 
years hence, to whom Newton and Leibniz, Napier and Biirgi, 
and scores of lesser contestants for individual fame will be semi- 
mythical figures as indistinct as Pythagoras. 

The harmonious geometry of Kepler’s laws challenged 
mathematical ingenuity to devise a hypothesis from which they 
could be deduced. Among others, Newton’s self-constituted rival 
and gadfly, the brilliantly original R. Hooke (1635-1703, 
English), had guessed and perhaps proved that Kepler’s laws 
implied an inverse-square law of attraction, but could not deter- 
mine the form of the orbit from this law. Newton, on being 
consulted in 1684, restored a proof, which he had discovered but 
mislaid, that the required orbit is an ellipse. This incident 
appears to have been the origin of the Principia. From his 
hypothesis of universal gravitation that any two particles o 


matter in the universe, of masses mi, at a distance d apart, 
attract one another with a force proportional to m^m^/d- (m 1 , 7zt», 
d, and the force being measured in the appropriate units), 
Newton deduced Kepler’s laws. 

The deduction would have been impossible without a 
rational dynamics. This had been provided by Galileo 57 and by 
Newton himself. Just as the Pythagoreans had reduced the 
intuitive perception of form to geometry, and the great geo- 
metrical astronomers from Eudoxus and Hipparchus to Coper- 
nicus and Kepler had reduced the motions of the planets to 
geometry', so Galileo undertook the reduction of all motion to 
mathematics. He advanced beyond his predecessors chiefly 
because lie aided reason by experiment, determining the facts in 
connection with falling bodies by accurate, controlled observa- 
tion before venturing to mathcmaticize. 

To some it seems incredible that any human being could 
ever have believed it possible to reason out the behavior of 
falling bodies without appeal to experiment. But one of the 
greatest intellects in history, Aristotle, had sufficient confidence 
in his logic to legislate for a universe which has but little respect 
for the unaided intellect. Others 1 * see nothing questionable in 
attempts such as Aristotle’s, substituting for the classical and 
medieval belief in Aristotelian logic an eager faith in the creative 
power of the intricate tautologies of mathematics. It may yet 
be too early to judge which side, if either, is right; but it is a fact 
that Galilean science, not Aristotelian logic and metaphysics, 
made our material civilization what it is. 

Whether or not the legendary' experiment in which Galileo 
confounded the Aristotelian scholastics by dropping shot from 
the Leaning Tower of Pisa ever took place, 55 Galileo knew by 
1591 that a one-pound shot and a ten-pound shot dropped 
simultaneously from the same height strike the ground simul- 
taneously. Experiments on motion down inclined planes gave 
him further data to be fitted into the mathematical theory' of 
motion he sought to construct. As tentative hypotheses were 
subjected to experimental verification, the cardinal definitions 
and postulates of dynamics began to emerge. 

In particular, Galileo mathcmaticizcd distance, time, veloc- 
ity, and acceleration into the scientific (experimentally measur- 
able) things they still are in classical dynamics. He sought to 
frame definitions that would respond to repeatable observations. 
He also understood an equivalent of Newton’s first postulate 


of motion — inertia: every body will continue in its state of rest 
or of uniform motion in a straight line, except in so far as it is 
compelled to change that state by impressed force. This postu- 
late contradicted the naive intuitions of Galileo’s predecessors 
and controverted the common sense of ages. 

Galileo also understood at least special cases of Newton’s 
second postulate: rate of change of momentum is proportional to 
the impressed force, and takes place in the direction in which the 
force acts. The mathematically important concept here is that 
of a rate; for rates are derivatives, and hence velocity, accelera- 
tion, and force are brought within range of the calculus. We have 
seen that Newton probably had velocity in mind when thinking 
about fluxions. 

“In the two plague years of 1665 and 1666,” as Newton 
states, he deduced from Kepler’s third law that “the forces 
which keep the Planets in their Orbs must be reciprocally as the 
squares of their distances from the centers about which they 
revolve: and therefore [I] compared the force requisite to keep 
the Moon in her Orb with the force of gravity at the surface of 
the Earth, and found them answer pretty nearly.” 30 

Further progress in the mathematical theory of gravitation 
was temporarily halted by Newton’s lack at the time of a 
theorem in the integral calculus: the gravitational attraction 
between two homogeneous spheres can be calculated as if the 
masses of the spheres were concentrated in their centers. Once 
this theorem is proved, the Newtonian law of universal gravita- 
tion is applicable. If there is one master key to dynamical 
astronomy, this is it. With this key Newton proceeded in 1685 
to unlock the heavens. He also, for the first time, gave a rational 
theory of the tides. 

The Newtonian celestial mechanics was the first of the great 
syntheses of natural phenomena. From its very nature, celestial 
mechanics without the dynamics of Galileo and Newton, or 
without the calculus of Newton and Leibniz, is unthinkable. 
The Galilean method in science was to provide the model for 
even more recondite mathematical syntheses, as in the theories 
of heat, light, sound, and electricity. But the modern scientific 
method, invented by Galileo and Newton, of welding experi- 
ment and mathematics into a single implement of discovery and 
exploration was to remain fundamentally the same as it was in 
Galileo’s Discorsi and Newton’s Principia. 


In a day when science is being discredited by messianic 
ignoramuses with enormous followings, it is well occasionally to 
recall the cliche, trite though it may be, that without this union 
of experiment and mathematics our civilization would not exist. 
Less trite is the more recent observation that because of this very 
union our civilization may cease to exist. And, while we arc 
facing facts, we note the opinion of many observers that ever 
since the days of Aquinas science has been feared or secretly 
hated by nine human beings out of every ten who have sufficient 
animation to hate or fear anything. Science has been grudgingly 
tolerated since the days of Galileo and Newton only because it 
has increased material wealth. If science dies, mathematics 
dies with it. 

To give an indication of how significantly dynamics and the 
Newtonian theory have influenced analysis, we may cite a few 
specific instances, some of which will be considered more fully 
in later chapters. Since the earth is not a sphere but a spheroid, 
its attraction on an exterior mass-particle cannot be calculated 
with the same precision as if its mass were concentrated at the 
center. When astronomy became more exact after Newton, the 
slight departure from perfect sphericity had to be included in the 
calculations, and this necessitated the invention of new func- 
tions, such as Legendre’s in potential theory. So rudimentary a 
dynamical problem as Galileo’s of the time of vibration of a 
simple pendulum of constant length leads at once in the general 
ease to an elliptic integral. Such integrals, by inversion, gen- 
erated the vast theory of doubly periodic functions. These in 
their turn were recognized in the late nineteenth century as but 
special eases of automorphic functions, whose theory still is far 
from complete. 

All of the earlier functions together suggested to Lagrange, 
Cauchy, and others in the late eighteenth and early nineteenth 
centuries general theories of functions, culminating in the theory 
of functions of a complex variable. Fourier’s analytical theory 
of heat (final form 1822), devised in the Galileo-Newton tradi- 
tion of controlled observation plus mathematics, is the ultimate 
source of much modern work in the theory of functions of a real 
variable and in the critical examination of the foundations of 
mathematics. Finally, the gravitational interactions of a system 
of mass-particles, in particular of three, generated the theory of 
perturbations and all its intricate analysis: and the problem of 


three bodies, partly topologized in the late nineteenth century, 
is the source of the modern theory of periodic orbits from which a 
qualitative, topologized dynamics is rapidly developing. 

Geometry also has enriched itself by successive alliances 
with mechanics. In the seventeenth century the astronomical 
need for the accurate measurement of time inspired Huygens to 
construct the first pendulum clock (1656). Incidentally, he was 
compelled to investigate the (small) oscillations of a compound 
pendulum, the first dynamical problem beyond the dynamics of 
particles to be discussed mathematically. From practical clock 
making Huygens tvas led to his great work 31 in horology (1673), 
in which he defined and investigated evolutes and involutes. 
The cycloid, sometimes called the Helen of geometry, partly on 
account of its graceful form and beautiful properties, figures 
prominently in this science. Huygens proved the remarkable 
theorem that the cycloid is the tautochrone. In more recent 
times, the four-dimensional geometry of Pliicker (nineteenth 
century) in which straight lines instead of points are taken as 
the irreducible elements of space, found a ready interpretation 
in the dynamics of rigid bodies. Conversely, this dynamics sug- 
gested much to be done in line geometry. But what is perhaps 
the greatest service a physical science has ever rendered geom- 
etry was the sudden acceleration imparted to differential 
geometry by Einstein’s general relativity and his relativistic 
theory of gravitation in the second decade of the twentieth 
century. In the curiously sanguine months after the buzzards of 
Versailles had completed their labors, it was frequently said 
that Einstein’s work would outlast the memory of the world war, 
as the science and mathematics of Archimedes has outlasted the 
Punic wars in the consciousness of all but professional his- 
torians. Twenty years later the human race had made a complete 
recovery from its attack of optimism. 

Here we leave the mathematics of the seventeenth century 
and commit ourselves to ■'che turbulent stream that gushed from 
that inexhaustible sou reek 


Extensions of Number 

In following the development of mathematics since the death 
of Newton (1727), we might start from any one of arithmetic, 
algebra, geometry, or applied mathematics. As arithmetic 
preceded the others in the historical order from Babylon to 
Gottingen, we shall discuss it first. Those who arc more inter- 
ested in one of the other topics may pass at once to it. 

The detailed growth of the number concept to be described 
in this and the following five chapters being quite intricate, we 
shall indicate first the principal features to be observed. 

Four critical periods 

About four centuries of generalisation, confused and hesi- 
tating at first, produced the number systems of analysis, algebra, 
mathematical physics, and the higher arithmetic of the twen- 
tieth century. The final gain left mathematics with three major 
acquisitions: the ordinary complex numbers of algebra and 
analysis, and their subclasses of algebraic integers; the hyper- 
complex number systems of algebra, geometry, and physics; 
the continuum of real numbers as it appears in the modern 
theories of functions of real and complete variables. The five 
periods of most radical change were the decade bisected by the 
year 1SD0, the late lS30*s and early lS-10's, the lS70*s, and the 
twenty years bisected by 1900. 

With the first period is associated one beginning of modern 
abstract arithmetic and algebra, in the use by Gauss (1 SOI ) of a 
particular equivalence relation, which he called congruence, to 
map an infinite class of integers on a finite subclass. The general 
method of mapping (homomorphism) implicit in this early 



work was not clearly formulated and isolated for independent 
study until the twentieth century, when it became basic in 
abstract algebra, topology, and elsewhere. 

In the 1830’s, the British algebraists clearly recognized the 
purely abstract and formal character of elementary algebra. This 
was followed, in the 1840’s, by Hamilton’s quaternions and 
the vastly more general algebras of Grassmann, from which the 
vector algebras of mathematical physics evolved. From the 
standpoint of pure mathematics, the lasting residue of this period 
was a widely generalized conception of number. 

The 1870’s saw the inception of the modern attack on the 
real number system in the work of Cantor, Dedekind, Meray, 
and Weierstrass. The outcome, in the late nineteenth century, 
was the arithmetization of analysis and the beginning of the 
modern critical movement. What now appears as the most 
enduring residue of this stormy period is the enormous expansion 
of mathematical logic during the first four decades of the twen- 
tieth century. 

The third period passed into the fourth about 1897 with the 
first appearance of the modern paradoxes of the infinite. The 
latter were largely responsible for the sudden growth of mathe- 
matical logic, which has reacted strongly on all mathematics 
and in particular on the number concept. 

We shall have occasion to refer frequently in anticipation to 
problems as yet unsettled concerning the nature of number and 
the continuum of real numbers. Because a problem defies solu- 
tion is no ground for believing it to be unsolvable. The outstand- 
ing obstacles that have hitherto blocked clear perception of the 
nature of number may be removed tomorrow. In any event, 
none of the unsolved problems of number has stopped progress in 
both pure and applied mathematics. On the contrary, in pure 
mathematics these unresolved difficulties have inspired much 
valuable work; and for applied mathematics, even the most 
serious doubts have as yet proved wholly irrelevant in obtaining 
scientific conclusions which can be checked against experience 
in the laboratory. 

Having followed the development of number in its purely 
mathematical aspects, we shall return in a later chapter to the 
impact of science on mathematics. It will be seen that applied 
mathematicians have been justified in their bold use of an 
analysis which may not yet meet all the demands of logica 



From time to time we shall cal! attention to points of special 
significance to be observed while following details. A general 
observation may be emphasized here. As mathematics passed the 
year 1800 and entered the recent period, there was a steady trend 
toward increasing abstractness and generality. By the middle of 
the nineteenth century, the spirit of mathematics had changed so 
profoundly that even the leading mathematicians of the eight- 
eenth century, could they have witnessed the outcome of half a 
century’s progress, would scarcely have recognized it as mathe- 
matics. The older point of view of course persisted, but it was no 
longer that of the men who were creating new mathematics. 
Another quarter of a century, and it had become almost a dis- 
grace for a first-rank mathematician to attack a special problem 
of the kind that would have engaged Euler in much of his work. 
Abstractness and generality, directed to the creation of universal 
methods and inclusive theories, became the order of the day. 
There had been one precedent in the eighteenth century' for such 
work, the dynamics of Lagrange. There is a second clue through 
the intricate development. This leads back to Pythagoras, and 
is so suggestive that we shall describe it next by itself in relation 
to the preceding sketch. 

The Pythagorean adventure 

Perhaps the feature of greatest general interest in the entire 
development is the wide departure from the Pythagorean pro- 
gram of basing all mathematics on the ‘natural* numbers 1, 2, 
3, . . . in the periods of intenscst creativity, and the final 
return to Pythagoras for a brief interval after the natural num- 
bers had been extended to meet the demands of analysis, geom- 
etry, physics, algebra, and the higher arithmetic. What was 
probably the golden age of the Pythagorean program lasted 
through the second half of the nineteenth century. Thereafter, 
the modern critical movement sought to base the natural num- 
bers, and hence all their acquired extensions, on mathematical 
logic. This later program was already strongly hinted in that 
of the nineteenth century', which had attempted to derive num- 
ber from the theory' of infinite classes. 

To be appreciated as the great adventure in thought that it 
is, this circling movement away from Py'thagoras and back to 
him, with the subsequent firing off at a tangent he was incapable 
of imagining, must be inspected in some detail. This we shall do 
in the following five chapters. We shall see modern mathematics 


as a whole becoming increasingly self-conscious and critical of 
its own naive behavior in the eighteenth century, and passing 
safely through adolescence in the decade 1820-30. Mathematics 
then became less interested in the uncritical analysis that 
produced surprisingly accurate results in the heavens with 
Laplace’s celestial mechanics (1799, 1802, 1805), and on earth 
in the intuitive calculations of Fourier’s analytic theory of heat 
conduction (1822). Most of the great mathematicians of the 
eighteenth and early nineteenth centuries were more like engi- 
neers than modern mathematicians in their thinking; a formula 
revealed in a flash of intuition, or hastily inferred from loose 
reasoning, was as good as any other provided it worked. Their 
formulas worked admirably. Gauss (1777-1855) was the first 
great mathematician to rebel successfully against intuition in 
analysis. Lagrange (1736-1813) had tried and failed. 

The focus of the last serious trouble was found most unex- 
pectedly in the speciously innocuous natural numbers 1, 2, 3, 
. . . that, since the days of Pythagoras, had been eagerly ac- 
cepted by mathematics as manna from heaven. Indeed L. 
Kronecker (1823-1891, German), himself a confirmed Pytha- 
gorean and one of the leading algebraists and arithmeticians 
of the nineteenth century, confidently asserted that “God made 
the integers; all the rest is the work of man.” By 1910, some of 
the more wary mathematicians were inclined to regard the 
natural numbers as the most effective net ever invented by the 
devil to snare unsuspecting men. Others, of a yet more mystical 
sect, maintained that the natural numbers have nothing super- 
natural of either kind about them, asserting that the ‘unending 
sequence’ 1, 2, 3, . . . is the one trustworthy ‘intuition’ vouch- 
safed to Rousseau’s natural man. The tribes of the Amazon 
Basin were not consulted. 

Clashes between these and other opposing factions of mathe- 
matical orthodoxy temporarily relegated the Pythagorean 
program to limbo shortly after 1900. All parties to the many- 
sided dispute united in torturing logic into new and fantastic 
shapes to make it reveal at last what meaning, if any, there may 
be in the natural numbers and in the dream of Pythagoras con- 
cerning them. Do these numbers tell the truth about mathe- 
matics and nature, or do they not, ‘truth’ being merely a 
self-consistent description ? If they do not, is it necessary for 
human needs that the mathematics constructed on the natural 
numbers be ‘true’ in this sense? Whatever may be the answer 



to the first question, that to the second seems to be an emphatic 
No. A vast amount of mathematical reasoning now known to be 
unsound led in the past to extremely useful consequences. 
However, our concern here is not with such profound questions, 
all of which may be meaningless, but with the technical mathe- 
matics which has bred these inquiries and many more like 
them. But we may note in passing that mathematics is not the 
static and lumpish graven image of changeless perfection that 
some adoring worshippers have proclaimed it to be. 

The extensions of the number system since the sixteenth 
century are one of the outstanding accessions of all mathematics. 
In the opinion of those competent to estimate the technical 
evidence, these extensions arc likely to be of value for many 
years to come. The ‘crisis’ of the early twentieth century in all 
mathematics, induced by an uncritical acceptance of the formal- 
ism that had generated the successive extensions of the number 
system, was precipitated by too bold a use of infinite classes in an 
attempt to be logically rigorous. Infinite classes penetrated the 
domain of number, as will appear later, from two diametrically 
opposite points, namely, the finite cardinal numbers of common 
arithmetic and the continuum of analysis. In the higher arith- 
metic also, Dcdckind’s generalization (about 1870) of the 
rational integers and their unique decomposition into primes to 
the algebraic integers with a corresponding unique decomposi- 
tion into prime ideals introduced dcnumerably infinite classes of 
algebraic integers. The concurrent entry of non-denumerably 
infinite classes of rational numbers came with the theories of 
Cantor and Dedckind, devised to provide the continuum of real 
numbers in analysis with a self-consistent foundation. Thus 
the central obstacle, the mathematics of the infinite, that had 
stopped Pythagoras halted his successors over two thousand 
years after he had become a legend. Eudoxus seemed to have 
discovered a way round the obstacle, or perhaps through the 
thickest part of it; and the builders of the modern continuum, 
following essentially the same path, sought only to clear it of 
obstructions and give it a firmer foundation. What at first looked 
like security appeared on closer inspection to be an illusion. The 
road had yet to be built. 

To anticipate the report in a later chapter and bring this 
forecast to a close, we recall the singular conversion of H. 
Poincare (1S54-19I2, I” rench) in 190S. This, in a way, sums up 
the progress of 2,300 years. At the close of the nineteenth cen- 


tury, Poincare was a major prophet of a self-confident mathe- 
matics. In 1900 he declared that all obscurity had at last been 
dispelled from the continuum of analysis by the nineteenth- 
century philosophies of number based on the theory of infinite 
classes {Mengenlehre). All mathematics, he declared, had finally 
been referred to the natural numbers and the syllogisms of tradi- 
tional logic; the Pythagorean dream had been realized. Hence- 
forth, reassured by Poincare, timid mathematicians might 
proceed boldly, confident that the foundation under their feet 
was absolutely sound. 

Eight stormy and eventful years changed the prophet’s 
vision: “Later generations will regard the Mengenlehre as a 
disease from which one has recovered.” 

Thirty years after Poincare’s somewhat caustic prognosis was 
pronounced, the theory from which mathematics was to have 
recovered was still flourishing. This of course disproves nothing; 
Euclid’s geometry lasted unmodified in the minds of generation 
after generation of mathematicians who for two thousand years 
believed it to be flawless. Both the prognosis and its possibly 
retarded realization are recalled here merely to exhibit in a just 
perspective the great acquisitions of number since the sixteenth 
century. If in its continual development mathematics seldom if 
ever attains a finality, the constant growth does mature some 
residue that persists. But it is idle to pretend that what was good 
enough for our fathers in mathematics is good enough for us, or 
to insist that what satisfies our generation must satisfy the next. 

Extension by inversion and formalism 

The earliest extensions of the system of natural numbers 
were the Babylonian and Egyptian fractions. These illustrate 
one prolific method of generating new numbers from those 
already accepted as understood, namely, inversion. To solve the 
problem ‘by what must 6 be multiplied to produce 2?’, a new 
kind of ‘number,’ the fraction must be invented. Here the 
direct operation is multiplication, and the inverse, division. The 
other pairs of elementary inverses are addition and subtraction; 
raising to powers and extracting roots. 

All of these elementary operations were known to the an- 
cients. The inverses, division and subtraction, of the rationa 
operations, multiplication and addition, necessitated the mven 
tion of common fractions and negative numbers; the operation 
inverse to powering was in part responsible for the invention o 



irrationals, including the pure imaginaries and the ordinary 
complex numbers. The solution of an algebraic equation, or of a 
system of such equations in several unknowns, can be restated 
as a problem in inversion with respect to iterations of addition 
and multiplication. Up to about 1840, algebraic equations were 
probably the most prolific source of extensions of the natural 

Considering the extensions up to and including the acquisi- 
tion of ordinary complex numbers, we shall take a point of view 
which may be indefensible historically but which can be justified 
mathematically: mere accidental encounters with, say, negative 
numbers do not constitute mathematical discover}'. Nor docs a 
rejection of imaginary roots of equations entitle anyone to 
priority in the invention of complex numbers. Until a conscious 
attempt was made to understand negative and complex numbers, 
and to state rules, however crude, for their use wherever they 
might occur, neither had any more right to be considered a 
mathematical entity than has an unconceivcd child to be con- 
sidered a human being. Mathematically, these numbers did 
not exist until the conditions indicated were met. 

Professional historians are in substantial agreement on the 
following details in the development of negatives. Diophantus, 
in the third century A.D., encountering —4 as the formal solution 
of a linear equation, rejected it as absurd. In the first third 
of the seventh century, Brahmagupta is said to have stated the 
rules of signs in multiplication; he discarded a negative root of 
a quadratic. The rules of signs became common in India after 
their restatement by Mahavira in the ninth century. AI- 
Khowarizmi, of about the same time, made no advance except 
that he appears to have exhibited a positive and a negative root 
for a quadratic without explicitly rejecting the negative. 

Of the Europeans, Fibonacci in the early thirteenth century 
rejected negative roots, but took a step forward when he inter- 
preted a negative number in a problem concerning money as a 
loss instead of a gain. It has been claimed that the Indians did 
likewise. L. Pacioli (1445 r-l 514, Tuscan) in the second half 
of the fifteenth century is credited w\tV a knowledge of the rule 
of signs on such evidence as (7 — 4) (4 — 2) = 3 X 2 = 6. 
M. Stifcl 1 (14S7I-1567, German) a fine algebraist for his time, 
called negative numbers absurd in the middle of the sixteenth 
century. Cardan, in his Ats map: a (1545), stated the rule ‘minus 
times minus gives plus’ as an independent proposition; he also 


is said to have recognized negative numbers as ‘existent,’ but 
on evidence which seems doubtful. In fact, he called negatives 

Bombelli in 1572 showed that he understood the rules of 
addition in such instances as m — n , where m, n are positive 
integers. Vieta, about the same time, rejected negative roots. 
Finally J. Hudde (1628-1704, Dutch) in 1659 used a letter to 
denote a positive or a negative number indifferently. As an 
historical curiosity, it may be mentioned that T. Harriot (1560- 
1621, English) w'as one of the first Europeans to duplicate the 
feat of the ancient Babylonians in permitting a negative number 
to function as one member of an equation. But he refused to 
admit negative roots. 

With one exception, all items in the foregoing list may be 
classified as partial extension by formalism. The extension was 
incomplete because no free use of negatives was made until the 
seventeenth century. The extension was formal because it had 
no basis other than the mechanical application of rules of cal- 
culation that were known to produce consistent results when 
applied to positive numbers, and were assumed to be legitimate 
in the manipulation of negatives. This unbased assumption was 
to be elevated in the 1830’s to the dignity of a general dogma in 
the notorious and discredited ‘principle of permanence of form.’ 
By the middle of the seventeenth century, untrammeled use of 
negatives had given mathematicians a pragmatic demonstration 
that the rules of common algebra lead to consistent results. But 
there was no attempt to go any deeper and put a substratum of 
postulates under the rickety formalism. 

The one glimmer of mathematical intelligence in the early 
history of negatives is the suggestion of Fibonacci that a negative 
sum of money may be interpreted as a loss. This appears to have 
been the ff:E&*<ri^p''fd\,fyard the second stage in the evolution of 
negatives, that of inte'rg-preting the results of formalism in terms 
of something -which is ■ accepted as consistent. It marks the 
beginning of two distinct but complementary philosophies of 
mathematics: the products of mathematical formalism arc to be 
admitted only if they c: x -.n be put in correspondence with some 
already established system accepted as self-consistent; all 
mathematics is a forim-alism without meaning beyond that 
inv ;ied by the postulates^ defining the formalism. For example, if 
Euclidean geometry is accepted as self-consistent, and if the 
formal algebraic operations with complex numbers can be inter- 



prctcd in terms of that geometry, the formalism of complex 
numbers is admissible. This is according to the first philosophy, 
which was that instinctively and subconsciously adopted by 
Fibonacci in his encounter with negatives. The second philosophy 
is illustrated by the rules of algebra in any modern elementary 
text, where a, b, c, • • • , *f-, X, — are displayed, and it is postu- 
lated that a = a, a + b = b- a, etc. 

Each philosophy has greatly enriched mathematics. The 
first, seeking interpretations, may be called synthetic; the 
second, beginning and ending in a formalism within its own 
postulated universe, 2 may be termed analytic. The designations 
arc merely for convenience, and are not intended to recall Kant’s 
terminology, although the parallel may be suggestive. The 
development of the number system is the record of a continual 
interplay between the synthetic and the analytic approaches. 
For applied mathematics, as in quaternions and the vector 
algebras that evolved from the geometrical interpretation of 
ordinary complex numbers, it is the synthetic philosophy that 
dominates; in pure mathematics the analytic alone is relevant. 

The history of mathematics holds no greater surprise than 
the fact that complex numbers were understood, both syn- 
thetically and analytically, before negative numbers. Accord- 
ingly, we shall retrace first the principal steps by which complex 
numbers arrived at mathematical maturity. The negatives will 
then enter incidentally. 

From manipulation to interpretation 

The early history of complex numbers is much like that of 
negatives, a record of blind manipulations unrelieved by any 
serious attempt at interpretation or understanding. The first 
clear recognition of imaginarics was Mahavira’s extremely 
intelligent remark in the ninth century that, in the nature of 
things, a negative number has no square root. He had mathe- 
matical insight enough to leave the matter there, and not to 
proceed to meaningless manipulations of unintelligible symbols. 
It is of more than historical interest that Cauchy 3 made the same 
observation a little less than a tho usan d years later (1847): 
“{'vc discard] the symbolic sign V— 1, which we repudiate 
completely, and which we may abandon without regret, because 
one docs not know what this alleged sign signifies, nor what 
meaning one should attribute to it.” These sentiments were the 


origin of Kronecker’s project in 1882-7 for a unified derivation 
of all extensions of the natural numbers. 

The next step forward after Mahavira was toward the 
analytic philosophy of number. Cardan in 1545 regarded imagi- 
naries as fictitious, but used them formally, as in the resolution 
of 40 into conjugate complex factors 5 ±y/ —15, without raising 
any question as to the legitimacy of the formalism. A more 
vicious species of pure formalism appeared in a totally unwar- 
ranted conjecture of A. Girard (?1590-?1633, Dutch). Having 
noticed that some equations of low degrees n have n real roots 
and that some quadratics have two imaginary roots, Girard 
inferred that any equation of degree n has n roots, supplying any 
awkward lack of real roots by guessing that the deficiency would 
be exactly met by complex roots. 

Leibniz 4 by 1676 had progressed no farther than Cardan. He 
gave a formal factorization of x 4 + a 4 , and succeeded in con- 
vincing himself that he had done something remarkable when he 
verified by actual substitution that Cardan’s solution of the 
general cubic in the irreducible case satisfies the equation. He 
was equally astonished by a similar verification of the expression 
of a special real radical as a sum of conjugate complexes. The 
truly astonishing thing historically about Leibniz’ performances 
with complex numbers is that less than three centuries ago one 
of the greatest mathematicians in history should have thought 
that any of these meaningless manipulations were mathematics, 
or that their outcome was more unexpected than is that of turn- 
ing a tumbler upside down twice in succession. That a mathe- 
matician, logician, and philosopher of the caliber of Leibniz could 
so delude himself, substantiates Gauss’ observation that “the 
true metaphysics of V - 1” is hard. It also suggests thatmathe- 
matics really has progressed since the ever-memorable seven- 
teenth century. 

In the eighteenth century, blind formalism at last produced 
a formula of the first magnitude. About 1710, R. Cotes (1682- 
1716, English), the man whose death moved Newton to lament, 
“If Cotes had lived, we might have known something,” stated 
an equivalent of the result usually called DeMoivre’s theorem 
in trigonometry. In current notation, i denoting 4 ® V~L Cotes 
formula 5 is i<f> = log e (cos <j> + i sin <£). DeMoivre’s theorem 
(1730), cos n<f> + i sin n f> = (cos <f> + i sin 4>) n [= J> f 1 an 
integer > 0, is an immediate formal consequence. Euler (1743, 
1748) extended the last to any n; he also gave the exponents 



forms of sin d>, cos 4 > — evident from Cotes’ result. Thus, by 1750 
trigonometry had become a province of analysis, and all that 
remained was to derive the analytic formulas with due attention 
to convergence and to create a self-consistent theory of complex 
numbers. The first desideratum was met in the third decade of 
the nineteenth century, by Cauchy; the second, in the last decade 
of the eighteenth century, by Wessel. So, after about a thou- 
sand years of meaningless mystery*, the so-called ‘imaginary’ 
numbers were incorporated into unmystical mathematics. 

Before proceeding to Wessel and his successors, we recall two 
noteworthy steps toward what has just been described. John 
Bernoulli observed the connection between inverse tangents and 
natural logarithms. This was in Cotes’ direction. Even more 
significant was the long stride toward a geometrical interpreta- 
tion of complex numbers taken in 1673 by J. Wallis (1616-1703, 
English), an original mathematician and at one time a fashion- 
able preacher. Wallis missed by a hairsbreadth the usual geo- 
metrical interpretation of complex numbers. But in mathematics 
a hair may be as thick as a ship’s cable; and Wallis is not usually- 
credited with Wesscl’s invention. In effect, Wallis represented 
the complex number x -j- iy by the point (.v, y) in the plane of 
Cartesian coordinates; what he missed was the use of the y-axis 
as the axis of imaginarics. 

It remained for a Norwegian surveyor, C. Wessel 0 (1745- 
1818), to take the final step and produce a consistent, useful 
interpretation of complex numbers in 1797. He modestly called 
his completely successful effort “an Attempt.” It fully explained 
what is customarily misnamed the Argand diagram in texts, and 
mapped the formal algebra of complex numbers on properties of 
the diagram. J. R. Argand (176S-1S22, French) independently 
arrived at similar conclusions in 1806. 

Wcssel’s decisive contribution suffered the misfortune of 
being published (1799) in a scholarly journal that mathemati- 
cians were not likely to read. A French translation in 1S97, 
exactly one hundred years after Wessel had communicated his 
paper to the Royal Danish Academy, secured its author what- 
ever reward there may be in posthumous fame. Thus a work 
that could have hastened the development of the number system 
might as well never have been written for all the influence it 
had; and it remained for the great authority of Gauss (1831) 
to get complex numbers accepted as respectable members of 
mathematical society. 


Two possible generalizations were suggested by WessePs 
interpretation. The geometry of complex numbers was obviously 
translatable into a description of rotations and dilatations in a 
plane. Were further extensions of the number system possible by 
which rotations in space of three dimensions could be described? 
Or were the complex numbers themselves adequate for the 
purpose? The answer to the first was to be affirmative, to the 
second, negative; but this could scarcely have been predicted 
in 1799 when Wessel’s interpretation was published. 

The geometrical approach was not the ‘natural’ way to the 
heart of the problem, although W. R. Hamilton (1805-1865, 
Irish) was to follow it successfully. But Hamilton, as will be seen, 
rested content with an algebra adapted to space of three dimen- 
sions. In mathematics, however, three is no more sacred than any 
other cardinal number, and the real problem was to extend 
complex numbers to ‘space’ of n dimensions. 

The Euclidean program 

Gauss took as the subject for his doctor’s dissertation (1799) 
a proof of the fundamental theorem of algebra: an algebraic 
equation has a root of the form a -f bi, a, b real. (For the precise 
statement of the theorem, we refer to any text on the theory of 
equations. The foregoing statement, like others in this account, 
is intended mereJy-*to recall the theorem.) After Girard’s con- 
jecture, there^had been attempts at proof, including essays by 
D’Alembert (1746) and Euler (1749). All were faulty, as were the 
first and fourth attempts (1799) by Gauss. 7 It may be stated in 
passing that the fundamental theorem in its classic form, as 
proved in the theory of functions of a complex variable, is no 
longer regarded as belonging to algebra. It is supplanted in 
modern algebra by a statement which is almost a triviality. 8 The 
basic ideas of the modern treatment go back to Galois (1811- 
1832), Dedekind (1831-191^, and Kronecker (1823-1891), not 
to Gauss. 

This first serious work of' the greatest mathematician since- 
Newton convinced him that 'a satisfactory theory of complex 
numbers had yet to be created. Unaware of Wessel’s work, Gauss 
himself arrived at a geometrical representation. 9 But the man 
who gave it as his mature opinion that “Mathematics is t e 
Queen of the Sciences, and Arithmetic the Queen of Matne- 
matics,” could not be satisfied with a helpful but irrelevant 



geometrical picture of tvhat, he believed, was purely a question 
of number. By 1811, Gauss had convinced himself that a 
‘formal’ treatment alone could provide a sound theory of com- 
plex numbers; and he came within an ace of committing himself 
to the mysterious principle of permanence which was to guide 
others to the desired end about a quarter of a century later. But 
in 1825 he confessed that “the true metaphysics of 'Sj—l” was 

By a formal treatment, Gauss meant the deduction of the 
properties of complex numbers from the accepted postulates of 
common arithmetic. He sought proofs in the manner of Euclid 
from definitions and explicit assumptions. We shall return later 
to the principle of permanence. 

What Gauss regarded as “the true metaphysics” of complex 
numbers was invented by him in 1831, six years before Hamilton 
communicated his independent discovery of the same method 
to the Royal Irish Academy. The ‘true metaphysics’ banished 
geometric intuition entirely, defining a -f- bi, a, k real, as the 
number-couple (a, b) subjected to postulates necessary and 
sufficient to yield the desired properties of complex numbers as 
given by algebraic manipulations. I'or example, equality 
(a, b) (c, d) is defined to mean a — c, b — d\ addition, 

(a, b) + (c, d), 

by definition, is {a + c, b T d); multiplication, [a, b) X (c, d) 
is ( ac — bd , ad + be). The mysterious i has vanished, and the 
algebra of complex numbers is replaced by what Dc Morgan 
and others called a “double algebra” of couples of real numbers 
a,b, c,d, . . . subject only to the accepted laws of arithmetic 
and common algebra, as a -f- b ~ b + a, ab = ha, 

a{b 4- c) — ab + ac, etc. 

The occasion for Gauss’ disclosing his anticipation of Hamil- 
ton’s method was a letter of 1837 from his old university friend 
W. Bolyai (1775-1856, Hungarian) in which Bolyai reproached 
Gauss for having propagated a geometrical theory of complex 
numbers. Bolyai argued that geometry has no place in the 
foundations of arithmetic, and that complex numbers should 
be referred to the real numbers whose arithmetic was assumed 
to be known. Gr.urs replied that he was of exactly the same 
epmion and that, in 1831, he had done what Bolyai demanded. 
He remained of ?hb opinion, and only five year" before his death 


emphasized that the “abstract,” postulational method is the 
desirable approach to complex numbers. This method is now 
fairly common in texts on college algebra. 

Anyone seeing the algebra or arithmetic of number-couples 
for the first time might be pardoned for thinking it a sly subter- 
fuge, or at best a beating of the imaginary devil round a sup- 
posedly real bush. Familiarity corrects misapprehension; and 
when the suggestive notation ( a , b ) for couples is extended to 
triples ( a , b, c ), and beyond, to sets of n ordered real numbers or 
elements of a field, with appropriately defined laws of addition 
and multiplication, the creative power of Hamilton’s simple 
invention becomes evident. Multiple algebra, with its innumer- 
able applications to the sciences, was already in sight when 
Hamilton replaced a + hi by ( a , b). He himself elaborated the 
algebra and geometry of number-quadruples ( a , b , c, d) in his 
quaternions; Grassmann almost simultaneously took a more 
general point of view and created the algebra of number-?z-ples, 
(ai, a 2 , , a n ). We shall resume this in a later chapter; for 

the moment, we follow the consequences of the resumption of 
the Euclidean methodology by Gauss and Hamilton. After about 
twenty-three centuries of sightless wandering, arithmeticians 
and algebraists opened their eyes and saw what Euclid had done: 
definitions, postulates, deduction, theorems. They then took a 
long stride ahead. 

It may have been clear to Euclid that his geometry was that 
of a postulated, ideal universe having no necessary connection 
with an intuitively perceived ‘real world’ of common experience; 
but if so, he did not convey the full import of his philosophy to 
his successors. Euclid’s geometry, or any other mathematical 
system constructed on the deductive pattern, is now almost 
universally regarded as a free and arbitrary creation of the 
mathematician constructing the system, whether the initial 
impulse came from experiences of the material world sublimated 
into abstractions, or whether it originated in formal extensions of 
algebraic symbolism, as in the passage from number-couples to 
ordered sets of n real numbers. The philosophy behind the 
Euclidean program as now conceived is analytic. 

It seems singularly appropriate that the conception of alge- 
bra as pure formalism should have first appeared in that country 
which above all others has revered Euclid. It was an Englishman, 
G. Peacock (1791-1858), at one time Lowndean professor in 
Cambridge University, later a Dean of Ely, who first 10 (18 , 


1845) perceived common algebra 11 as an abstract hypothctico- 
dcductivc science of the Euclidean pattern. 

Peacock was not an ‘important’ mathematician in the ac- 
cepted sense of wide reputation; so possibly the following is a 
just estimate of his place in mathematics: “He was one of the 
prime movers in all mathematical reforms in England during the 
first half of the 19th century, although contributing no original 
work of particular value.” 12 He was merely one of the first to 
revolutionize the whole conception of algebra and general 

The Euclidean program advocated by Peacock was developed 
by the British school, notably by D. F. Gregory (1813-1844, 
Scotch), and A. Dc Morgan; but it did not become widely known 
until H. Hankcl (1839-1873) in 1867 expounded it with insight 
and massive German thoroughness. Hankcl also reformulated 
the principle of permanence of formal operations, which had 
been stated in less comprehensive terms by Peacock: “Equal 
expressions couched in the general terms of universal arithmetic 
arc to remain equal if the letters cease to denote simple ‘quan- 
tities,’ and hence also if the interpretation of the operations is 
altered.” For example, ah — ba is to remain valid when a, b are 

It is difficult to see what the principle means, or what possible 
value it could have even as a heuristic guide. If taken at what 
appears to be its face value, it would seem to forbid ah — —ba, 
one of the most suggestive breaches of elemental-}- mathematical 
etiquette ever imagined, as every student of physics knows from 
his vector analysis. As a parting tribute to the discredited prin- 
ciple of permanence, we note that since 2 X 3 = 3 X 2,itfol!ows 
at once from the principle that \^2 X V3 — X \5. But 
the necessity for proving such simple statements as the last was 
one of the spurs that induced Dedekind in the 1870’s to create 
his theory of the real number system. According to that peerless 
extender of the natural numbers, “Whatever is provable, should 
not be believed in science without proof.” 13 

The device of number-couples, invented to exorcise the 
imaginary and reduce the theory of complex numbers to that of 
pairs of real numbers, also banished rational fractions and 
negative numbers. Thus, for negatives, — is replaced by 
[«:, ir, -4 «], where v: is an arbitrary positive number; zero is 
{«:, and r. is -f n, »:]. As the details are available in stand- 
ard texts, we pass on. The last and most difficult step in reducing 


all ‘number’ to the natural numbers 1, 2, 3, . . . concerned the 
real irrationals. This arithmetized analysis. 

Between the final step and the formalizing of algebra and 
arithmetic by Peacock, De Morgan, Hamilton, and others, the 
natural numbers were vastly extended in another direction, that 
of algebraic numbers, beginning with Gauss in 1831 and continu- 
ing into the twentieth century. Concurrently, the generalizations 
of number-couples to multiple algebra were developed. Another 
type of arithmetization, originating in 1801 in the work of Gauss 
and reaching one of its climaxes in Kronecker’s work of 1882-7, 
incidentally provided another means of reducing all numbers to 
the natural numbers. This will be described in the following 

The Euclidean program, which would ultimately reduce all 
mathematics to a pure formalism, had its opponents as well as 
its partisans in the nineteenth century as it has had since. To 
illustrate the ironies of prophecy, we recall the vigorous attack 
delivered in 1882 by a distinguished analyst, P. du Bois-Rey- 
mond (1831-1889, German), whose penetrating researches 
contributed much to the progress of analysis in its second heroic 
age — that of Newton and Leibniz being the first — in the nine- 
teenth century. The program of formalism, du Bois-Reymond 
declared with considerable passion, would replace mathematics 
by “a mere play with symbols, in which arbitrary meanings 
would be attached to the signs as if they were the pieces on a 
chessboard or playing cards.” He went on to prophesy that such 
a ‘meaningless’ outcome would fritter out in barren efforts and 
be the death of mathematics as Gauss had pictured the Queen 
of the Sciences. Since 1920, mathematics for one highly produc- 
tive school has become exactly what the prophet feared it might. 
Those who call themselves formalists revel in their endless game 
of chess and exult that it has no meaning whatever beyond the 
rules of the game. Ultimate realities and eternal truths, at least 
in mathematics and science, suffered an eclipse in the twentieth 

Thus has ended one quest after the meaning of number; and 
this conclusion, disconcerting to some, was reached by following 
the same road that Euclid took. It was D. Hilbert’s (1862-1943, 
German) close scrutiny of the postulates of elementary geometry, 
in an endeavor to put a solid foundation under that venerable i 
somewhat palsied science, that led him to a similar inspection o 
the bases of common arithmetic. Addressing the second inter- 



national congress of mathematicians in 1900, Hilbert observed 15 
that the noncontradiction of the postulates of geometry is dem- 
onstrated by constructing a suitable domain of numbers such 
that, to the geometrical postulates, there correspond analogous 
relations between the numbers of the domain. Consequently, any 
contradiction in the conclusions drawn from the geometrical 
postulates would necessarily be recognizable in the arithmetic 
of the domain. Thus the self-consistency (noncontradiction) of 
the postulates of geometry is referred to the self-consistency of 
the postulates of arithmetic. Consequently Hilbert emphasized 
as one of the outstanding unsolved problems of mathematics in 
1900 the proof that, by proceeding from the postulates of arith- 
metic, it is impossible to reach contradictory results by means of 
a finite number of logical deductions. The problem was still open 
in 19-15. Attempts to solve this seemingly elementary problem 
were in part responsible for the formalistic — chess-playing — 
school of mathematical philosophy led by Hilbert, the foremost 
mathematician of his generation. 

Enough has been said to indicate the fundamental impor- 
tance of the natural numbers for all mathematics, not merely for 
arithmetic and its algebraic extensions, and to suggest that 
Euclid’s methodology is as vital in modern mathematics as it 
was in ancient. It will be well before proceeding to further 
extensions of number to cast up the account of Pythagoras with 
mathematics thus far. 

Pythagoras to 1900 

Glancing back over the confused effort to incorporate imagi- 
naries with the reals in one self-consistent number system, we 
note the curious fluctuations in mathematical creed accompany- 
ing the struggle. The abrupt check experienced by the Pytha- 
goreans in their encounter with irrationals practically abolished 
mensuration in orthodox Greek mathematics, and the investiga- 
tion of number independently of its geometrical representation 
all but ceased, Euclid’s summary* of arithmetica being a partial 
exception. The academic Greek mathematicians were at ease 
with number only when it was geometrized into ‘magnitude’ — 
a vague concept whose tenabilitv they seem never to have 
questioned. Thus number was supposed to be apprehensible 
through form, the opposite of what the Pythagoreans first held 
and of what a majority of mathematicians have believed since 


The diagrams of Wessel and Argand were an unseasonable 
reversion to pre-Cartesian mathematics. Exactly nothing is 
provable by the geometrical representation of complex numbers 
unless it be assumed that the underlying geometry is founded 
consistently. Gauss also, we have seen, at first sought to ‘justify’ 
imaginaries geometrically, but later decided that this was a 
mistake. In all these earlier developments, geometry was ac- 
cepted without question as the irreversible court of last appeal. 
But with increasing sophistication, it was perceived that the 
geometrical justification is merely disguised arithmetic, real 
numbers entering with the coordinates of a point in the plane of 
complex numbers. The geometrical interpretation is thus left 
without a foundation until the real number system is firmly 
based in self-consistency. Going deeper, Hilbert in 1899 resumed 
the Pythagorean program for all geometry, referring form to 
number and demanding a proof of noncontradiction for the real 
number system, or even for its subset of rational integers. 

Hamilton also was a Pythagorean in his escape from geom- 
etry into number-couples. His was the more suggestive method 
for future extensions to hypercomplex numbers. But he was less 
critical than Hilbert, in that he took the self-consistency of the 
real number system for granted. 

The modern attack, as in abstract algebra, attempts to strip 
Hamilton’s number-couples ( a , b ) of all arithmetical connota- 
tions by postulating that the ‘coordinates’ a, b are defined by the 
postulates of an abstract field. The last vestige of number as the 
Pythagoreans conceived it has been sublimated from the ‘mean- 
ingless marks’ a, b and their equally ‘meaningless rules of com- 
bination.’ But the problem of proving that the rules will never 
produce a contradiction is not eliminated by manipulating a set 
of postulates which, by assumption, completely define the 
mathematical system deducible from them. 

In escaping from form to number, and back from number to 
form, thence again to number, and finally into complete ab- 
straction, mathematicians from Pythagoras to Hilbert have 
sought to validate their creations by deductive reasoning. 
Hilbert was the first to recognize the futility of such vacillation 
until deductive reasoning as applied in all mathematics should 
itself be shown to be incapable of producing contradictions. This 
put the whole Pythagorean program on trial; for the cardina 
hypothesis of the Pythagoreans assumed that number and form 
may be consistently described by deductive reasoning. The fina 



question, then, is, How far can mathematical deduction be 
trusted if it is not to produce contradictions such as l X is equal 
to B, and A is not equal to B' ? This is debated in the symbolic 
language of reasoning foreseen by Leibniz. Some of the conclu- 
sions will be noted in the last chapter. 

For the moment it is sufficient to observe that the ability 
of modern mathematics to discuss such questions profitably is 
one of its titles to superiority in power over its predecessors. 
And, whatever the outcome is to be, the admitted utility' of 
complex numbers, whether conceived as affixes of points in a 
plane or as number-couples, in both pure and applied mathe- 
matics, will doubtless remain substantially unaffected. 


Toward Mathematical Structure 


Three new approaches to number, in 1801 and in the 1830’s, 
were to hint at the general concept of mathematical structure 
and reveal unsuspected horizons in the whole of mathematics. 
That* of 1801 was the concept of congruence, introduced by Gauss 
in what many consider his masterpiece, the Disjuisitiones arith- 
meticae, published when its author was twenty-four. To this and 
the revolutionary work (1830-2) of E. Galois (1811-1832, 
French) in the theory of algebraic equations can be traced the 
partial execution of L. Kronecker’s (1823-1891, German) revolu- 
tionary program in the 1880’s for basing all mathematics on the 
natural numbers. 

The same sources are one origin of the modern abstract 
development of algebraic and geometric theories, in which the 
structure of mathematical systems 1 is the subject of investiga- 
tion, and it is sought to obtain the interrelations of the mathe- 
matical objects concerned with a minimum of calculation. 
‘Structure’ may be thought of for the present in any of its intui- 
tive meanings; it was precisely defined in 1910 by the 
mathematical logicians. It might be compared to morphology 
and comparative anatomy. We shall approach mathematical 
structure through the union effected in the nineteenth century 
between algebra and arithmetic. 

Abstraction and the recent period 

From the standpoint of mathematics as a whole, the method- 
ology of deliberate generalization and abstraction, culminating 
in the twentieth century in a rapidly growing mathematics o 



structure, is doubtless the most significant contribution of all 
the successive attempts to extend the number concept. But at 
ever}' stage of the progression from the natural numbers 1, 2, 
3, ... to other types of numbers, each of several fields of 
mathematics adjacent to arithmetic v.-as broadened and enriched. 

New acquisitions in other fields reacted reciprocally on 
arithmetic. For example, the first satisfactory theory of ordinary 
complex numbers to become widely known was that of Gauss 
(1831), devised to provide a concise solution for a special problem 
in diophantinc analysis: If p, q arc primes, what conditions must 
p, q satisfy in order that at least one of the equations 
ari = qy + p, z* = pee + shall be solvable in integers x, y, z, 
r r? The theory of complex numbers necessitated a radical revi- 
sion and generalization of the concept of arithmetical divisibility, 
which in turn suggested a reformulation of certain parts (inter- 
sections of varieties) of algebraic geometry. The latter in its 
turn was partly responsible for further generalizations (modular 
systems) in the algebraic arithmetic — or arithmetical algebra — 
of the twentieth century. 

The like may be observed in the creation of the numerous 
vector algebras invented during and after the 1840's for applica- 
tion to the physical sciences. The first of these evolved directly 
from the vectorial interpretation of ordinary complex numbers. 
The extension in the lS40’s of vector algebra in a plane to space 
of more than two dimensions was one origin of the hypercomplex 
number systems of algebra, and these again supplied arithmetic 
with new species of integers. The development of the correspond- 
ing arithmetic in turn reacted, particularly in the twentieth 
century, on the algebra in which it had originated. It would 
seem to be incorrect, therefore, to say that any one division of 
mathematics was alone responsible for the steady progression 
since 1800 from the special and detailed to the abstract and 
general. The forward movement was universal, and each major 
advance in one department induced progress in others. 

In following this development, one misapprehension above ail 
others that might be possible is to be particularly guarded 
against. Those who are not mathematicians by trade arc some- 
times inclined to confuse generality with vagueness, and abstrac- 
tion with emptiness. The exact opposite is the ease in the 
mathematical generalizations and abstractions with which we 
shall be concerned here. Each, on appropriate and definitely 
prescribed specialization, yielded the specific instances from 


which it had evolved. The theory of hypercomplex numbers for 
example, contains as a mere detail that of ordinary complex 
numbers; and once the general theory of hypercomplex number 
systems has been elaborated, the special theory of ordinary 
complex numbers follows automatically. Moreover, each gen- 
eralization gives in addition a whole universe of mathematical 
facts distinct from those in the special instances from which the 
generalization proceeded. 

It was remarked in the Prospectus that the separation of all 
mathematical history into a remote period, to 1637, a middle 
1638-1801, and a recent, 1801 — , distinguishes three well-marked 
epochs in the development of mathematics. Following the rapid 
growth of arithmetic and algebra, we shall see that in the passage 
from the middle period to the recent there was a profound change 
in the quality of mathematical thought and its objectives. This 
change is most simply observed, perhaps, in the evolution of the 
number concept. It is the item of greatest interest to be noticed 
in this chapter and the following. Geometry might have been 
considered instead of arithmetic to exhibit the same change. But 
as the transformed algebra and arithmetic played important 
parts in the expansion of geometry, it seems more natural to 
consider them first. It is to be borne in mind, however, that 
while arithmetic and algebra were being transformed into shapes 
the mathematicians of the eighteenth century would not have 
recognized as mathematics, geometry and analysis were under- 
going corresponding transformations. 


The abstract approach of the 1830’s to algebra parallels 
the epochal advance in geometry made simultaneously with the 
publication in 1829 of N. I. Lobachewsky’s (1793-1856, Russian) 
non-Euclidean geometry. This also stems from 1800 or earlier 
in the preparatory work of Gauss and others. As this properly 
belongs to geometry, it will be discussed in that connection. The 
relevant detail here is that geometers and algebraists perceived 
almost simultaneously that mathematical systems are not 
supernaturally imposed on human beings from without, but are 
free creations of imaginative mathematicians. Lobachewskys 
new geometry was the earliest mathematical system to be 
recognized as such a free creation. It provided the first proof 
of the complete independence of a particular postulate (Euclid s 
postulate of parallels) in a system which tradition and common 


sense had agreed must contain that postulate. The significance 
of this radical step in methodology was only slowly appreciated; 
and it would seem that the almost simultaneous advance of 
algebra and arithmetic in a parallel direction was more directly 
responsible than geometry for the modern abstract view of 

The explicit recognition, by the British school in the 1830’s, 
of common algebra as a purely formal mathematical system 
shortly led to a revolution in arithmetic and algebra of signifi- 
cance comparable to that precipitated by non-EucIidean 

Hamilton’s rejection (1843) of the commutative ‘law’ (postu- 
late) of multiplication, in his invention of quaternions, opened 
the gates to a flood of algebras, in which one after another of the 
supposedly immutable ‘laws’ of rational arithmetic and common 
algebra was either modified or discarded outright as too restric- 
tive. By 1850 it was clear to a majority of creative mathemati- 
cians that none of the postulates of common algebra, which up 
to 1843 had been thought necessary for the self-consistency of 
symbolic reasoning, was any more a necessity for a noncontradic- 
tory algebra than is Euclid’s parallel postulate for a self-con- 
sistent elementary geometry. To the astonishment of some, it 
was found that the modified algebras, such as Hamilton’s 
quaternions, were adaptable to mechanics, geometry, and mathe- 
matical physics. The dead hand of authoritative tradition had 
been brushed aside; mathematics was free. As G. Cantor (1845- 
1918, Germany), one of the boldest extenders of the number 
concept, was to say about three-quarters of a century' later, “the 
essence of mathematics is its freedom.” No mathematician, not 
even Gauss, could have conceived such a thought in 1801. The 
accomplished facts of the revolutions in geometry and algebra of 
the 1830’s and lS40’s made freedom conceivable. 

From supernaluralism to naturalism 

The change during the nineteenth century from what may be 
called Platonic supcrnaturalism to modern naturalism in mathe- 
matics is reflected in three aphorisms, the first of which may' 
have expressed the Greek reverence for synthetic geometry; the 
second, the early ninctccnth-ccntury worship of arithmetic and 
nnalv sis; and the third, the final admission that mathematics is 
made by men. The second and third, following the example of 
the first, were phrased in classic Greek. 


Plato is said to have asserted that “God ever geometries’’- 
C. G. J. Jacobi (1804-1851, German), a great arithmetician and 
analyst, declared that “God ever arithmetizes”; while J. W. R. 
Dedekind (1831-1916, German), first and last an arithmetician 
wrote as the motto for his famous essay on the nature of number 
{Was sind und was sollen die Zahlen?, 1888), “Man ever 

A proposition less open to objection than the last today 
would be “Man attempted to arithmetize during the second half 
of the nineteenth century, and came to grief so doing early in the 
twentieth.” Although this is less elegant than the original, it is 
closer to historical fact. Nevertheless, man’s failure as yet to com- 
plete the Pythagorean program of arithmetizing mathematics 
and the universe has been, and is, a potent stimulant to the con- 
tinual creation of new and interesting or useful mathematics. 

During the nineteenth century the physical sciences were 
given to solving the universe by dissolving it in vast generaliza- 
tions distilled from inadequate data. So powerful were some of 
these solvents that they dissolved themselves. Having learned by 
disconcerting experience that the universe is not to be solved 
between breakfast and lunch, the physical sciences took a more 
modest view of their function, and in the early twentieth century, 
after critical introspection, contented themselves with consistent 
descriptions intelligible to instructed human beings. Universe- 
solving went out of fashion temporarily about the time of the 
first world war. 

Mathematics in the meantime was experiencing similar diffi- 
culties in its abortive struggle to comprehend its own vast empire 
in the all-inclusive generalization of Pythagoras. What is to be 
the outcome is not yet predictable. But two things may be 
reasonably conjectured. 

The Pythagorean project of deriving all mathematics from 
number will continue for many years to suggest new accessions 
to mathematics, and it will remain essentially as the Pythag- 
oreans imagitfpd it in their unmystical moments. If form, for 
example, is to be better described by our successors in terms of 
something other than number, we have no inkling at present of 
what that something may be, unless it be symbolic logic or 
analysis situs, themselves in partial process of arithmetization. 

We may also conjecture that mathematics, like the physica 
sciences, will take a less inflated view of itself as a result of its 
critical self-analysis. There will be less mysticism in the mathe- 


matics of the future than there has been in that of the past, and 
fewer grandiose claims to immortality and eternal truth. Mathe- 
matics will become less self-conscious, less introspectively 
critical, and more boldly creative. It will resign its soul to the 
metaphysicians for such tortures as they may choose to inflict, 
feeling nothing; for it will continue to serve with its living body 
the purposes of the men who create it to meet human needs 
rather than to be the plaything of sterile philosophies. The 
very implements of torture — symbolic logic, for example — were 
devised in mathematics itself as by-products of more immedi- 
ately useful inventions. 

It is the latter that are of vital significance in a scientific 
civilization; the by-products too often carry with them the cold 
smell of a mildewed scholasticism. The spirit of the Middle 
Ages, which the successors of Galileo and Newton imagined 
they had laid forever in science, stirs again in the twentieth- 
century disputes concerning the nature and meaning of number. 
Ignoring these for the present, we shall continue with the more 
profitable arithmetic of 1801 which, a century later, was to 
deliquesce in metaphysics. But history will compel us to return 
to these disputes. Our attitude in what follows will be that of 
Molierc’s despised ‘‘average sensual man,” who seeks through 
science merely to make life less barbarous for himself and his 
fellows, and who is content to leave what professed humanists 
call “the really important questions” to God and the 

Congruence from 1801 to 1887 

‘Congruence,’ like ‘analysis,’ ‘formal,’ ‘ideal,’ ‘functional,’ 
‘analytic,’ ‘normal,’ ‘conjugate,’ ‘modulus,’ ‘integral,’ and a 
dozen others, is one of those overworked technical terms in 
mathematics which appear to have been invented to confuse the 
uninitiated by a multitude of meanings having no connection 
with one another. Congruences in higher geometry, as in con- 
gruences of lines or circles, arc unrelated to congruence in 
elementary geometry, as in congruent triangles; and congruence 
in arithmetic, with which we arc concerned here, has nothing in 
common with congruence elsewhere. Nor arc the ideal elements 
of projective geometry significantly connected with ideals as in 
arithmetic and algebra. 

Gauss in 1S01 defined two rational integers a , b to be con- 
gruent with respect to the rational integer modulus m if, and only 


if, a , b leave the same remainder on division by m; and he 
expressed this by writing a s= b mod m. Otherwise stated if 
a 2 = b mod m, then a — b (or b — a) is a multiple of m, and 
conversely; and x s 0 mod m asserts that # is exactly divisible 
by m. 

This simple but profound invention is one of the finest illus- 
trations of Laplace’s remark that a well-devised notation is 
sometimes half the battle in mathematics. Writing ‘x is divisible 
by ml as x = 0 mod m at once suggested to Gauss extremely 
fruitful analogies between algebraic equations and arithmetical 
divisibility. The last is one of the central and most elusive con- 
cepts of all arithmetic. It is not this technical aspect of con- 
gruence, however, that is of primary importance for our 
immediate purpose, but another, of far deeper significance, which 
was perceived only by the successors of Gauss. If Gauss did 
foresee this, he seems to have left no record of the fact. 

To bring out the point, of the first importance for an under- 
standing of modern mathematical thought, we must return for a 
moment to a hypothetical prehistory, long before Pythagoras 
and even prior to Sargon. Abstraction, to judge by the behavior 
of contemporary primitives, is not by any means ‘natural’ to 
Rousseau’s carefree savage. Numbers first were nouns as con- 
crete as father and mother, an early instance, perhaps, of ‘one,’ 

No trace survives of the actual passage from concreteness to 
abstractness, when ‘two’ was realized as applicable to a couple of 
parents, a stick and a stone, or any other of its innumerable 
manifestations; and we can only imagine the dismay of human 
beings when they were first overwhelmed by the appalling 
revelation that the natural numbers have no end. Traces of the 
attempt to cope with that first deluge of knowledge survive in 
the symbolism, meaningless to us, of number mysticism. Sym- 
pathetically viewed, all that prehistoric nonsense was the 
outcome of men’s first groping efforts to regiment the generative 
freedom, c n into n + 1,’ of the numbers in the unending sequence 
1, 2, 3, + 1, .... If only some finite restraint 

could dominate the endless generations of numbers, they would 
be less terrifying. 

It must have given the mind that first perceived that odd 
and ‘even’ suffice to comprehend all the natural numbers a sense 
of almost supernatural power. The endless sequence after all 
was no more mysterious than humanity itself, which could be 
subsumed under ‘male’ and ‘female.’ Accordingly, the mat e- 


matically useful separation of the natural numbers into only 
two classes was made more concretely satisfying to the primitive 
mind by calling odd numbers male and even numbers female. 
Arithmetic and numerology thereafter flourished together in 
happy and fruitful symbiosis. But however nonsensical the 
numcrological fruits of that early union may now appear, its 
occasion was the urge to comprehend an infinite totality in finite 
terms, and hence to bring the infinite within the grasp of a finite 

Gaussian congruence has proved the most fruitful of all 
classifications of the rational integers 0, +1, ±2, ±3, . . . into 
a finite number of classes, as may be appreciated on inspecting 
any elementary text on the theory of numbers. What Gauss 
could not have foreseen was that his invention of mapping one 
assemblage, finite or infinite, of individuals on another, by 
classifying the individuals in the first set with respect to some 
relation having the abstract properties of reflexiveness, sym- 
metry, and transitivity, shared by his relation of congruence, 
was to become a guiding principle to the structure of algebraic 
theories. With the gradual evolution of this and similar ideas, 
mathematics transcended the Pythagorean dream and, as in 
the theories of groups, fields, point sets, symbolic logic, etc., 
escaped from the natural numbers into a domain where number 
is irrelevant and the structure of relations is the subject of 

The concepts mentioned above being fundamental in modern 
mathematics, we shall recall their definitions. A relation, denoted 
by is said to be binary with respect to the members a, b, 
c, ... of a given class of things (which need not be ‘numbers’ 
of any kind) if a ~ b is either true or false for any a, b in the 
class. If a ~a for every a in the class, ~ is reflexive; if a ~ b 
implies b ~ a, ~ is symmetric; and finally, if a ~ b and b ~ c 
together imply a ~ c, ~ is transitive. A relation such as ~ is 
called an ‘equivalence relation’ for the given class. Equality, =, 
is a simple instance of If in, a,b,c, . . . arc rational integers, 
and in pi 0, and if a ~ b is interpreted as a ~ b mod in, it is 
easily verified that this Gaussian congruence is an equivalence 
relation. Further, congruence is preserved under addition and 
multiplication: if .v e= a mod in, y — b mod m, then .v -f- y -- 
a + b mod in, and .yy ab mod m. 

Any equivalence relation separates its class, whether finite 
or infinite, into subclasses, all those members, and only those, 


of the whole class that are equivalent to a particular member 
(and hence by transitivity to one another) being included in a 
particular subclass. Any member of a subclass may be taken as 
representing the entire subclass. Congruence with respect to the 
positive integer modulus m separates all the rational integers 
into precisely m classes, whose representatives may be taken as 
0, 1, 2, . . . , m — 1. Congruence is a typical example, and 
historically the first, of the modern methodology of mapping an 
infinite totality on a comprehensible finite set. The theory of 
arithmetical congruence as developed by Gauss and his succes- 
sors belongs to the higher arithmetic, and will be noted in that 
connection. Our present interest in congruence is in yet another 
direction, of deeper significance than the technical applications 
to arithmetic for mathematical thought as a whole. 

In the preceding chapte r we noted Cauchy’s objections 
(1847) to the symbol i ( = V~l)- An immediate extension of 
Gaussian congruence to congruences between polynomials in 
one variable (more properly, 'indeterminate’) x provided Cauchy 
with the escape into the illusory ‘reality’ he so ardently desired. 

F, = jf) A r x m ~ r , and M, - ^ B s x n ~\ 

r «= 0 «"0 

are polynomials, with in ^ n and A 0 B 0 ^ 0, there is exactly one 
polynomial R of degree n — 1, and exactly one polynomial 
Q, such that Bo'~ n+i F = QM + R. Cauchy wrote the particular 
case of this in which B 0 = 1 as a congruence, F = R mod M, and 
imitated the Gaussian theory in his easy development of such 
congruences for polynomials. 

For the particular modulus x 2 + 1( = M), Cauchy found that 
his ‘residues’ R had all the formal properties of complex numbers, 
his ‘x’ taking the place of ‘t.’ He was thus enabled to construct a 
wholly ‘real’ algebra abstractly identical with (having the same 
structure as) that of complex numbers. A moment’s reflection 
will show why his ingenious device succeeded. It offered an 
alternative to Hamilton’s number-couples. 

It seems rather surprising that Cauchy, having gone so far, 
should not have continued to the expulsion of negatives from his 
paradise of ‘real,’ ‘existent’ numbers, for they surely are as 
‘unreal’ and as ‘inexistent’ as i to the Pythagorean mind. Natu- 
rally enough, the Cauchy who in 1821 had given the first satis- 
factory definitions of limits and continuity in the calcu us 


noticed nothing demanding reform in 1847 in the continuum of 
real numbers with their non-denumcrabie infinity of irrationals. 
A thoroughgoing Pythagorean would have expelled the real 
irrationals along with i. 

Cauchy extended his invention to what he called algebraic 
keys; but, as these were very special instances 2 of some of the 
algebras 5 already implicit in the work (1S44) of H. G. Grass- 
mann (1809-1877, German), they have rather missed fire. The 
prolific Cauchy passed on to new creations more conformable to 
his passion for analysis. His ingenious suggestion went unnoticed 
for forty years, when (1S87) it reappeared, greatly amplified, in 
the arithmetical program of Kronecker.' 1 

Here at last was the modern Pythagoras. Gauss is said to 
have ascribed an ‘external reality 5 to ‘space 5 and ‘time, 5 while 
reserving for number the ideal purity of a ‘creation of the mind.’ 
Kronecker denied this philosophy, insisting that geometry and 
mechanics are expressible wholly in terms of relations between 
numbers, and by numbers he meant the positive integers 1, 2, 
3, . . . . Thus, for him, the continuities ‘space 5 and ‘time, 5 fused 
in the concepts of kinematics, had meaning only in terms of the 
ineradicable discontinuities of these same God-given natural 
numbers. Continuity had no meaning; all was discrete. 

To show how his subversive program might be carried out, 
Kronecker expelled negative numbers by means of congruences 
to the modulus j + 1, precisely as Cauchy had banished imagi- 
narics a -f- hi with his modulus f 2 + 1. Since only the natural 
numbers existed for Kronecker, he exorcised rational fractions 
by a similar magic, introducing (in effect) a new symbol, or 
‘indeterminate, 5 for each objectionable fraction. To dissipate 
-J, for example, it sufficed to use congru ence s to the compound 
modulus •}/; -f- 3 j. An irrational, say could be dispensed 

with by a new indeterminate t and an additional modulus l- -j- 2. 
Arithmetic, algebra, and analysis began to grow somewhat 
complicated. But that was beside the point. 

In the work cited and in an extensive earlier memoir (1882), 
Kronecker outlined in some detail how the program of Pythag- 
oras could be realized in modern mathematics. Whether such a 
project was worth doing is irrelevant. Kronecker was interested 
primarily in showing that the Pythagorean vision could be 
materialized. Provided that were once demonstrated, careless 
mortals presumably were to be permitted to use negatives and 
irrationals in the customary manner and in the usual notations, 


on the understanding, however, that they admitted their work- 
able mathematics to be merely a convenient shorthand for the 
only true mathematics, that of Kronecker’s modular systems. 

It would be interesting to know what Gauss would have 
thought of this devastating outcome of his simple device of 
writing ‘n is divisible by m’ as n — 0 mod m. Eminent mathe- 
maticians have called it anything from anarchy to hocus-pocus. 
Kronecker, however, might have recompensed himself at the 
expense of the nineteenth-century analysts, had he lived to 
participate in the debates of the twentieth century on the con- 
sistency of classical analysis. For few in the 1940’s would have 
written with the resolute conservatism of E. W. Hobson (1856— 
1933, English) in 1921: 

Kronecker’s ideal . . . that every theorem in analysis shall be stated 
as a relation between positive integral numbers only, . . . , if it were possible 
to attain it, would amount to a reversal of the actual historical course which 
the science has pursued; for all actual progress has depended upon successive 
generalizations of the notion of number, although these generalizations are 
now regarded as ultimately dependent on the whole number for their founda- 
tion. The abandonment of the inestimable advantages of the formal use in 
Analysis of the extensions of the notion of number could only be characterized 
as a species of Mathematical Nihilism. 

Apart from its Pythagoreanism, Kronecker’s effort left a 
useful residue, his theory of modular systems. This provides an 
alternative approach to algebraic numbers, Dedekind’s being 
that usually followed. 

One of the elementary by-products of Kronecker’s algebra 
(1882) provided a rigorous theory of elimination for systems of 
polynomials in any number of variables. This rendered obsolete 
many unsatisfactory attempts, particularly by algebraic geome- 
ters, to give sound proofs for the speciously simple formalism of 
such methods as grow out of J. J. Sylvester’s (1814—1897, 
English) of 1840, and E. Bezout’s (1730-1783, French) of 1764. 
The latter was also invented independently by L. Euler (1707- 
1783, Swiss). The usual textbook discussion is still in the spirit 
of 1764, although there are honorable exceptions. 

The same cynical fate awaited Kronecker’s reduction of all 
mathematics to the natural numbers that seems sooner or later 
to nullify all human attempts to solve the universe at one stroke. 
It does not appear to have occurred to him that the natural 
numbers themselves might some day be put on trial as he had 
tried all other numbers and found them wanting in meaning. 


Any savage might have suggested such a possibility; but it 
remained for the mathematical logicians of the early twentieth 
century to demonstrate the possibility up to the hilt and beyond. 

A -period of transition 

We must briefiv indicate the involuntary participation of 
E. Galois (1S11-1S32, French) and N. H. Abel (1S02-1S29, 
Norwegian) in the development of Kronecker' s Pythagoreanism. 
Galois himself adhered to no such creed. Nor did Abel. But it 
was in the attempt to understand and elucidate the Galois theory 
of equations, left (1S32) by its young author in a rather frag- 
mentary and unapproachable condition, that Kronecker ac- 
quired some of his skill. Both Kronecker and Dedekind, two of 
the founders (E. E. Rummer [1810-1893, German] being a 
third) of the theory of algebraic numbers, were inspired partly 
by their scrutiny of the Galois theory to begin their own revolu- 
tionary work in algebra and arithmetic. Kronecker also began 
some of his researches in the arithmetization of algebra with a 
profound study of abelian equations. 

Galois and Abel mark the beginning of one modern approach 
to algebra. The transition from highly finished individual 
theorems to abstract and widely inclusive theories is plainly 
evident in the algebra of Gauss contrasted with that of Abel and 
Galois. The like is seen in other fields, as will appear presently. 
This transition took place about 1S30, contemporaneously with 
the abstract approach of the British algebraists. 

Younger than Gauss by thirty-four years, and dying twenty- 
three years before him, Galois now, curiously enough, seems 
more modern than Gauss. A single example will suffice to sub- 
stantiate the radical distinction between the two minds. 

The occasion for Gauss’ making mathematics his lifework was 
his spectacular discovert- at the ace of nineteen concerning the 
construction of regular polygons by means of straightedge and 
compass alone. Gauss proved that such a construction is possible 
if, and only if, the polygon has ?; sides where r. is an integer 
of the form 2* Pjpr s *5 0, in which pi. p*. . . . , p. arc r 

different primes, each of the form a power of 2 plus 1. The alge- 
braic equivalent of this theorem, concerning binomial equations, 
is partly developed in the seventh and last section of the Dis- arithmeticae. This particular work marks the end of 

its era in mathematical outlook. 


These two liberators of algebra are among the nineteenth 
century’s major mathematical prophets. Both were richly gifted 
in many things besides mathematics. Hamilton at the age of 
thirteen was an accomplished classicist and a proficient linguist 
in the oriental as well as the European tongues; Grassmann 
was a profound scholar of Sanskrit. At twenty-seven Hamilton 
was famous as the result of his mathematical prediction of con- 
ical refraction, a deduction from his comprehensive theory of 
systems of rays in optics; and by thirty he had practically com- 
pleted his fundamental work in dynamics, an advance beyond 
Lagrange comparable to Lagrange’s beyond Euler. At the age of 
thirty-eight (1843) he overcame the difficulty which had pre- 
vented him from extending the algebra of coplanar vectors to a 
theory of vectors and rotations in space of three dimensons. Ide 
discovered that the commutative law of multiplication is not 
necessary for a self-consistent algebra. Thereafter Hamilton’s 
scientific life was devoted to the elaboration of his theory of 
quaternions, in the mistaken hope that the new algebra would 
prove the most useful addition to mathematics after the dif- 
ferential and integral calculus. 

Plonors were showered on Hamilton; none fell on the less 
fortunate Grassmann. Neither had a particularly happy life. 
Hamilton was afflicted by domestic troubles and personal weak- 
nesses; Grassmann supported himself, his wife, and nine children 
by elementary teaching, a profession for which he was eminently 
unsuited. A steadfastly pious man, Grassman trusted that if his 
contemporaries failed to reward his signal merits, the Lord 
would. He never complained of the torments he endured from the 
young savages he was meagcrJy paid to civilize. His avocations 
were his life — the Sanskrit classics, philosophy, phonetics, 
harmony, philology, physics, theology, politics being among the 
extraordinary miscellany. But with the possible exception of 
theology, Grassmann’s creation of “a new branch of mathe- 
matics in 1840-4 gave him the most abiding satisfaction. Here 
his inventive imagination and his perverse originality had free 
play. His theory of extension ( Ausdehr.ungslchre ), in which 
PInmilton’s quaternions are a potential detail, was first published 
in 1844, about a year after Hamilton had found the clue to his 
problem of rotations in the equations = i 1 ~ h- — ijk — — 1 
defining the quaternion units :, /, 

It has often been observed that it is not healthful for a 
mathematician to be a philosopher. Whether or not this is a 
general theorem, it was certainly true in the unfortunate Grass- 


mann’s case. Endowing his theory with the utmost generality it 
could support, he all but smothered it in philosophical abstrac- 
tions. This was one of the greater tragedies of mathematics 
Gauss looked the Ausdehnungslehre over, and blessed it with his 
qualified approval. It was partly in the same directon, he said 
as he himself had taken almost half a century earlier. But it was 
too philosophical with its “peculiar terminology” even for 
Gauss, himself no inconsiderable amateur of philosophy. 

Gauss in the meantime had recorded his own independent 
discovery of Hamilton’s quaternions. In a brief abstract which 
he never published, 6 ascribed to the year 1819, Gauss wrote out 
the fundamental equations of what he called mutations in space, 
essentially quaternions. 

Grassmann continued his efforts to gain recognition for his 
own incomparably more general theory. Eighteen years (1862) 
after the first publication of his book, he brought out a com- 
pletely revised, greatly amplified, and somewhat less incom- 
prehensible version. 5 But a mathematician who has once been 
seriously called a philosopher might as well have been hanged 
for all the hearing he is likely to get from his fellow technicians. 
The second edition followed the first into temporary oblivion, 
Grassmann abandoned mathematics. The scope of his theory was 
perhaps not fully appreciated until the twentieth century. As 
one implicit detail, Grassmann’s work included the algebra of 
the tensor calculus that became widely known only after its 
application (1915-16) in general relativity. 

The central difficulty that had blocked Hamilton in his 
attempt to create an algebra of vectors in space of three dimen- 
sions was the commutative law of * multiplication. His own 
graphic account of how he saw his wa) round the obstruction in 
a flash of certainty after much fruitless work being readily acces- 
sible, we need not repeat it here. But it is well worth thoughtful 
consideration by all students, especially by those who imagine 
that mathematical inventions fall into people’s laps from 

Before Hamilton succeeded, able men had failed to find the 
clue to a consistent algebra of rotations and vectors in space. 
For one, A. F. Mobius (1790-1868, German), who in 1823 had 
been a pupil 7 of Gauss, took a considerable step toward the 
desired algebra of four fundamental units in his barycentnc 
calculus of 1827, a work which Gauss complimented as being 
composed in the true mathematical spirit. But Mobius was 


balked by the commutative law of multiplication, which he 
lacked the daring to reject. However, his new algorithm was of 
importance in the development of analytic projective geometry, 
particularly in the use of homogeneous equations, and he was an 
independent discoverer of the geometric principle of duality.’' 
So his effort was anything but wasted. 

The four fundamental units 1, f, /, k of Hamilton’s quarter- 
nions a + hi -f- cj + dk , (a, b, c, d real numbers), do for rotations 
and stretches in space what 1, i do for the like in a plane. But 
whereas multiplication of complex numbers is commutative, that 
of quaternions is not. Familiar as mathematicians are today with 
swarms of algebras in which the postulates of common algebra 
arc severally violated, they can still appreciate the magnitude of 
Hamilton’s success when, in a flash, he transcended the tradition 
of centuries. His insight is comparable to that of the founders 
of non-Euclidcan geometry, or to that of the arithmeticians who 
restored the fundamental law of arithmetic to the seemingly 
lawless algebraic integers. 

It is radical departures from traditional orthodoxy such as 
these that carry' mathematics forward what seems like a century 
or more at one stride. The painstaking, detailed cultivation of a 
newly discovered territory is necessary if it is to be fruitful; but 
such work can be well done by' mere competence, while radically' 
new discovery (or invention) is possible only' to men who may 
imagine they arc conservatives, but who at heart arc rebels. 
Their boldness may cost them their scientific reputations or the 
comforts of a decent livelihood; for the way' of the transgressor — 
who may be only a harmless innovator with the courage to step 
out in front of the rabble of respectable mediocrity — is some- 
times as hard in science as it is elsewhere. Grassmann paid for 
his rashness with eighteen years of obscurity and final scientific 
extinction for the remainder of his life. Gauss, long in possession 
of non-Euclidcan geometry, preferred his peace of mind to what 
he called “the clamor of the Boeotians,” and kept his treasure 
to himself. Hamilton, having won an imperishable success in 
optics and dynamics, courted the indifference of his contem- 
poraries when he devoted all of his superb talents to quaternions 
and, during his lifetime, acquired exactly one competent disciple 
in algebra. P. G. Tnit (1831-1901, Scotch) gave up his all in 
mathematics to follow quaternions. 

Ten years (1853) after his initial discovery, Hamilton pub- 
lished his Lectures or. quaUrr.ior.s (64 736 Ixxii pages), in 


which he showed the utility of quaternions in geometry and 
spherical trigonometry. But the geometry was Euclidean and of 
three dimensions. The massive Elements of quaternions (lvii -J- 
76 2 closely printed pages) followed in 1866, the year after Hamil- 
ton’s death. If anything could have convinced geometers and 
physicists that quaternions were the master key to geometry, 
mechanics, and mathematical physics that Hamilton antici- 
pated, his Elements should have done so. Literally hundreds of 
applications to these subjects were made by Hamilton in this 
elaborate work, which he considered his masterpiece. 

Many reasons have been suggested for the failure of qua- 
ternions to fulfill Hamilton’s expectations. A sufficient explana- 
tion, which includes many of the others, is that the calculus of 
quaternions was simply too hard for the busy scientists whom 
Hamilton would have helped. It took too long to master the 
tricks. But the possibility of an algebra specifically adapted to 
Newtonian mechanics and some parts of mathematical physics 
had been more than merely suggested, and it was reasonably 
certain that such an algebra would be forthcoming when the 
need for it became acute. Whatever form the desired algebra 
might assume, it was also a fair guess that it would follow the 
example of quaternions and reject the commutative law of 

On the long view, then, the permanent residue of Hamilton’s 
tremendous labor was the demonstrated existence of a self- 
consistent algebra in which the commutative law of multi- 
plication does not hold. This in turn, like the invention of 
non-Euclidean geometry, encouraged mathematicians to break 
the iron law of custom elsewhere and to create new mathematics 
in defiance of venerated traditions. A striking instance, which 
was to prove of cardinal importance in the development of 
algebra and the number system, was the construction of algebras 
in which ab = 0 without either a or b being zero, or in which 
a n ^ 0 (n = 0, 1, . . .i , w), but a m+1 = 0. A simple instance of 
the former occurs in hyoolean algebra (belonging to the algebra 
of logic), in which the ''fact stated is the symbolic expression of 
Aristotle’s law of contradiction. Linear associative algebra 
furnishes any desired number of algebras containing 'divisors 
of zero’ — such as a, b described above — also any number of 
algebras of the second species. The origin of all these modifica- 
tions of common algebra is in the work of Hamilton and Grass- 
mann in the 1840’s. 


Grassmann’s outlook was much broader than Hamilton’s. 
To appreciate how much broader it was, we must remember that 
in 1844, when Grassmann published his first Ausdehnungslehrc, 
‘space,’ for all but A. Cayley (1S21-1895, English), was still 
imprisoned in Euclid’s three dimensions. Cayley’s sketch of a 
geometry of n dimensions is dated 1843; it could not possibly 
have influenced Grassmann’s theory of ‘extended magnitude,’ 
which also can be phrased in the language of ?;-dimensional 
space. A ‘real’ space or ‘manifold’ of n dimensions is the set, or 
class, of all ordered n-plcs (.Vi, A'*, . . . , x„) of n real numbers 
a'i, a*j, . . . , A'„, each of which ranges over a prescribed class of 
real numbers. It is sufficient for purposes of illustration to let 
each of x x , a';, . . . , ,v„ range independently over all real num- 
bers. The class of all (.Vt, x s , . . . , x„) is also called an Jt-dimen- 
sional real number manifold. 

In effect, Grassmann associated with (x x , a-j, . . . , x„) the 
hypercomplex number AVi 4" A';C: 4* * * • 4* x„e n , where e x , 
c», . . . , e„ arc the fundamental units of the algebra, which he 
proceeded to construct, of such hypcrcomplcx numbers. Two 
such numbers, xh*i + ' ‘ + x„e n and y\C\ + • • • + arc 
defined to be equal if, and only if, a*i = yi, . . . , x n = y„. 

Addition was defined by 

(.ViC) 4- • • * + AVn) + (yid + • • • y„fn) 

- {xj + yj)o + • • • + (•’••„ 4~ yn)fn, 

of which an instance is common vector addition if n — 2 or if 
?: = 3. The various special kinds of multiplication that can be 
defined at will give the general algebra its chief interest. 

To demand a definition of multiplication without stating 
what properties the product is to have is meaningless. If, for 
example, the associative law a(bc) = ( ab)c is to be preserved, 
this is equivalent to imposing certain conditions on the funda- 
mental units Ci, ... , if either of the distributive laws, 
a{b 4- c) — ab 4" nc, ( b 4* c)a — ba 4 - ca, is to hold, this must 
be expressed in terms of relations between rj and 
similarly for the commutative law of multiplication, ab —ba. 
Thinking partly in terms of geometrical imagery, Grassmann 
defined several types of multiplication. In particular, multiplying 
out (ujCs 4 - • - • 4 - a n fn)(bjfi 4- ' ‘ • 4- £ r .<v>)* mid assuming 

that the ‘coordinates’ a a b u . . . , b n commute with 

the units e Xi . . . , e n . so that — ajjtc x r t , etc., Grass- 

mann called e x e u e x e z , c z c h . . . , <r n _ 5 r P ., <v.r n _ j, in the distributed 


product + <Zi& 2 ^i ^2 + azbie<ie\ + • • • units of the sec- 

ond order; and he first imposed conditions on these new units. 
For example, the product of a x e r + • • • and Vi + ‘ • was 

called an inner product if e r e, = 1 or 0 according as r = s or 
r ^ s; and an outer product if e T e s = —e s e r , forr, s = 1, . . . n% 
From these two kinds of products, Grassmann constructed others 
for more than two factors. For example, if e r je s denotes the inner 
product of e r , e s , and [ e r e s ] the outer product, there are the 
possibilities [e r \e s ]e t , e r \[e s e t ], among others, for the definition of 
products of three factors. A type of particular importance is that 
in which each of the n 2 products e r e s is a linear homogeneous 
function of the fundamental units e h ... , e n , and multiplica- 
tion is postulated to be associative. The linear associative alge- 
bras_of B. Peirce (1809-1880, U.S.A.), developed in the 1860’s 
but first printed in 1881, are of this type. 9 

A third type of product, called ‘open’ or ‘indeterminate,’ 
was to prove of central importance in the creation (1881-84) of 
a practical vector analysis by J. W. Gibbs (1839-1903, U.S.A.). 
The modern name for such a product is a matrix. 10 Gibbs, one 
of the most powerful mathematical physicists of the nineteenth 
century, was perhaps better qualified than Grassmann or Hamil- 
ton to sense the kind of algebra that would appeal to students of 
the physical sciences. His most original mathematical contribu- 
tions in this direction were in dyadics and the linear vector 

These hints must suffice to suggest that as early as 1844, 
Grassmann was in possession of an extensive theory capable of 
almost endless developments by specialization in various direc- 
tions. As elaborated by its creator, this theory of ‘extended 
magnitudes’ might be interpreted as a greatly generalized vector 
analysis for space of n dimensions. It incidentally accomplished 
for any finite number of dimensions what Hamilton’s quaternions 
were designed to do for Euclidean space of three dimensions. 
We have already noted that Grassmann’s algebra includes 
quaternions as a very special case. As a general kind of algebra it 
also includes the theories of determinants, matrices, and tensor 
algebra. In short, Grassmann’s theory of 1844-62 was anywhere 
from ten to fifty years ahead of its epoch. 

Our present interest in Grassmann’s work is the wide general- 
ization it afforded of complex numbers X\ + ix-i as number - 
couples (xi, X 2 ) to hypercomplex numbers {x\, . . . , #„). We 
must now relate this extension of the number concept to another, 


made explicitly in 1858 by Cayley but already implicit in the 
work of Grassmann, namely, matrices. The elements of the 
theory of matrices are now included in the usual college course in 
algebra; and since their appearance (1925) in the quantum 
theory, matrices have become familiar to mathematical 

The invention of matrices illustrates once more the power and 
suggestiveness of a well-devised notation; it also exemplifies the 
fact, which some mathematicians are reluctant to admit, that a 
trivial notational device may be the germ of a vast theory having 
innumerable applications. Cayley himself told Tait 11 in 189-1 
what led him to matrices. “I certainly did not get the notion of a 
matrix in any way through quaternions: it was either directly 
from that of a determinant; or as a convenient mode of expres- 
sion of the equations 

x' = ax + by 

y’ = ex + dy.” 

Symbolizing this linear transform 

variables by the square array ^ 

meats,’ Cayley was led to his algebra of matrices of n- elements 
by the properties of linear homogeneous transformations of n 
independent variables. 

Behind this invention there is a relevant bit of history. 
Cayley had shown (1858) that quaternions can be represented as 

matrices a with a, l, c, d certain complex numbers. To Tait, 

the pugnacious champion of quaternions ever since he had 
elected himself Hamilton’s disciple in 185-1, 11 this discover)' of 
Cayley's was conclusive evidence that Cayley had been inspired 
to matrices by his master’s quaternions. Because the multiplica- 
tion of matrices is in general not commutative, and since the 
like is true of quaternions, therefore, etc. This illustrates the 
unreliability of circumstantial evidence in mathematics as else- 
where. But for Cayley's testimony, critics might even now be 
asserting that Hamilton anticipated Cayley in the invention of 
matrices, or at least that Cayley got the notion of a matrix from 

Applications, or developments, of these extensions of number 
followed two main directions. The first, in the geometrical 
tradition of Hamilton and Grassmann, led to the extremely 

ation on two independent 
^ of its coefficients or ‘cjc- 


useful vector algebras of classical mechanics and mathematical 
physics, and later to the tensor algebra and calculus of relativity 
with its modifications and generalizations in modern differentia) 
geometry, also to the matrix mechanics of the quantum theory. 
The second, in the arithmetical spirit of Gauss, guided in part 
by the abstract algebraic outlook of Galois, led to a partial but 
extensive arithmetization of algebra. The course of both was 
highly intricate and blocked by innumerable details, many of 
which still promise to be of some enduring significance. But to 
see the principal trends at all, special and strictly limited devel- 
opments must be ignored, for the present at least; and we shall 
attend only to the shortest paths from the past to the gains just 

From vectors to tensors 

The line of descent of vector algebra in general is fairly clear. 
The composition of velocities or of forces in the corresponding 
parallelogram laws suggested the addition of ‘directed magni- 
tudes.’ Wessel’s or Argand’s diagram for depicting complex 
numbers was equally suggestive visually, geometrically, and 
kinematically. Hamilton and De Morgan’s ‘double algebra’ of 
number-couples, replacing that of complex numbers, naturally 
suggested a generalizatidn>.to number-triples, -quadruples, and 
so on. As we have seen, the central difficulty was the purely 
algebraic obstacle of commutative multiplication. Thus, in at 
least the early stages, geometrical and mechanical intuition 
shared about equally with formal algebra in the creation of a 
workable mathematics of vectors. 

The famous Treatise on natural philosophy (1879) of Thomson 
and Tait offered a magnificent opportunity to display the power 
of quaternions as an implement of exposition and research in 
mechanics. Tait exhorted Thomson to repent of his Cartesian 
sins and embrace the true faith of quaternions. But W. Thomson 
(Lord Kelvin, 1824-1907, Scotch), declaring that Hamilton’s 
good mathematics had ended with the masterpieces on optics 
and dynamics, hardened his heart and persisted in his iniquitous 
coordinates. The great opportunity was missed. 

Tait had a somewhat better success with J. C. Maxwell 
(1831-1879, Scotch). In his epoch-making Treatise on electricity 
and magnetism (1873, Art. 11), Maxwell made a slightly damning 
concession: “I am convinced . . . that the introduction of the 
ideas, as distinguished from the operations and methods of 


Quaternions, will be of great use . . . especially in electro- 
dynamics ...” And, with one exception, Maxwell studiously 
avoided quaternions. The exception (Art, 618) is a summary in 
quaternion notation of the electromagnetic equations. No use is 
made of this summary. But Maxwell did use “the ideas,” not of 
quaternions, but of his own conception of vector analysis. His 
convergence is the negative of the divergence in use today, and 
he introduced (Art. 25) what is now called the curl of a vector. 
These innovations have lasted. 

The most profitable departure from quaternionic orthodoxy 
was that of J. W. Gibbs in his vector analysis of the ISSO’s. This 
will be noted presently. The next was by O. Heaviside (1850- 
1925, English), in his profoundly individualistic Electromeig7ietic 
theory of 1893. In a chapter of 173 pages, Heaviside elaborated 
Ins own vector notation. His methods resembled those of Gibbs; 
but of Gibbs’ notation, Heaviside confessed, “I do not like it.” 
Germany provided the next (1S97) considerable variation on the 
now familiar theme, in A. Foppel’s Gcovictrie der JVirb elj elder — 
geometry of vortex fields. By 1900 the contest between rival 
claimants to physical favor had narrowed down in English- 
speaking countries to Gibbs versus Heaviside. Quaternions 
appeared to have been knocked out. Tait, their most formidable 
champion, died in 1901; the vector analysis of Gibbs or some 
modification of it prevailed in the U.S.A. 

Much of this tortuous development was enlivened by one of 
the most spirited mathematical controversies of modern times. 
Unlike the numerous squabbles over priority, the quaternions- 
vcrsus-vcctors war was refreshingly scientific. The casus belli 
was a purely mathematical difference of opinion: were quater- 
nions a good medicine for applied mathematics, or was some one 
of several diluted substitutes a better? The uninitiated might 
think that so abstract a bone of contention would provoke only 
dry academic discourse, with at worst an occasional growl of 
dissent. It did nothing of the kind. The language of the dis- 
putants even bordered on quite un-Victorian indelicacy at times, 
as when Hamilton’s devoted Tait ,: in 1890 called the vector 
analysis of Gibbs “a sort of hermaphrodite monster, com- 
pounded of the notations of Hamilton and Grassmann.” That 
was Scotch and Irish against American. Gibbs, being a New 
Englander to the marrow and a confirmed bachelor cherished 
only by his married sister, was but slightly acquainted with the 
inexhaustible resources of the American language. Tait got away 


with his abnormal physiology, but Gibbs got the better of the 
mathematical argument. 

Frenchmen, Germans, and Italians, urging their respective 
substitutes for quaternions, added to the din. By the second 
decade of the twentieth century there was a babel of conflicting 
vector algebras, each fluently spoken only by its inventor and 
his few chosen disciples. If, at any time in the brawling half- 
century after 1862, the bickering sects had stopped quarreling 
for half an hour to listen attentively to what Grassmann was 
doing his philosophical best to tell them, the noisy battle would 
have ended as abruptly as a thunderclap. Such, at any rate 
seems to have been the opinion of Gibbs. In retrospect, the fifty- 
year war between quaternions and its rivals for scientific favor 
appears as an interminable sequence of duels fought with stuffed 
clubs in a vacuum over nothing. 

The disputes ceased to have any but a mathematically trivial 
significance almost as soon as they began. As Gibbs 10 emphasized 
in 1886, in his account of the development of multiple algebra, 
the mathematical root of the matter is in Grassmann’s indeter- 
minate product, that is, in the theory of matrices. Gibbs also 
remarked the superior generality of Grassmann’s many possible 
kinds of product in multiple algebra over the unique product 
insisted upon by Hamilton: 

Given only the purely formal law of the distributive character of multipli- 
cation — that is sufficient for the foundation of a science. Nor will such a science 
be merely a pastime for an ingenious mind. It will serve a thousand purposes 
in the formation of particular algebras. Perhaps we shall find that in the most 
important cases the particular algebra is little more than an application or 
interpretation of the general. 

The whole of Gibbs’ judicial and profound evaluation (1886) 
of multiple algebra in relation to its applications might be 
studied with profit at any time by those interested in the con- 
tinued improvement of applied algebra. Vector analysis and even 
the infinitely more inclusive Ausdehnungsleliren of Grassmann 
are after all only provinces, although highly cultivated ones, of 
algebra, which itself is but a territory of modern mathematics. 
Those interested in the advancement of mathematics, rather 
than in the perpetuation of individuals as dictators of provinces, 
will not be dismayed when particular theories to which they may 
be personally attached are supplanted by others. Obsolescence 
is a necessary adjunct of progress; and any effort such as Tait s 
to keep quaternions unsullied and perpetually fresh is likely to be 


as futile as an attempt to stop the earth in its orbit. The vector 
analysis of Gibbs gradually displaced quaternions as a practical 
applied algebra in spite of the utmost efforts of the quaternion- 
ists; and after 1916 it seemed that the several special brands of 
vector analysis were about to be supplanted in their turn by the 
tensor algebra and analysis that became popular in 1915-16 
with the advent of general relativity. 

As in the struggle of vector analysis against quaternions, the 
advance to tensors generated its own opposition. Vector analysis, 
like some human beings, needed above all else to be delivered 
from the good intentions of its partisan friends. Progress here, as 
elsewhere in the past of mathematics, appeared to be possible 
only when all the friends and former pupils of some great and 
justly famous master should have died. Then only might it be 
possible to see the mathematics rather than the man. 

Such retardations due to misdirected enthusiasm are frequent 
enough in mathematics. The master founds a ‘school’; the pupils, 
remembering perhaps among other things an encouraging pat on 
the head from their first competent teacher, graduate into a 
world that docs not stop dead no matter who dies, to keep on 
repeating for the rest of their lives the only lesson they ever 
really learned. The school itself expires, leaving its useful con- 
tribution encrusted with an accumulation of artificially stimu- 
lated growths that must be cut away before the creative idea of 
the originator can begin to live and function freely. Aware of 
these possibilities, some mathematicians, including one of the 
first rank, have refrained from propagandizing their own ideas 
or those of their teacher, and have made no attempt to gather a 
following of bigoted disciples. Kronecker took pride in the fact 
that he had never tried to found a school or to acquire a host of 
students. He believed, as did Gibbs, that “the world is too large, 
and the current of modern thought is too broad, to be confined 
by the ipse dixit even of a Hamilton.” 

There seems to be but little doubt that applied algebra was 
held back by the partisans of jealous schools. The road to unity 
can be traced back from about 1940, when the rudiments of the 
tensor calculus had become fairly common in undergraduate 
instruction, to Grassmann’s n-dimensional manifolds of 1844. 
Three dimensions arc inadequate for modern physics, or even 
for classical mechanics with its generalized coordinates. G. F. B. 
Ricmann (1S26-1866, German) in 1854 took the next long step 
forward after Grassmann when he introduced Gaussian (in- 


trinsic) coordinates and made 72 -dimensional manifolds basic in 
his revolutionary work on the foundations of geometry. Another 
work of Riemann’s, published after his death, contained what 
is now known as the Riemann-ChristofFel tensor in the rela- 
tivistic theory of gravitation. Riemann encountered this tensor 
in a problem on the conduction of heat. E. B. Christoffel (1829— 
1900, German) was the nest to make significant progress toward 
a general tensor calculus, in his work of 1869 on the transforma- 
tion (equivalence) of quadratic differential forms. Finally, in 
the 1880’s, the Italian geometer M. M. G. Ricci combined and 
added to all the work of his predecessors. The result, published 
in 1888, 13 was the tensor calculus. Thus the mathematical 
machinery demanded by the theory of general relativity was 
available a year after the Michelson-Morley experiment, which 
was partly responsible for the special theory of relativity in 
1905; without the tensor calculus the general theory' of 1915-16 
would have been impossible. The above assertion about the 
Michelson-Morley experiment does not imply that Einstein 
was motivated by' the experiment in his construction of special 
relativity. In fact he has stated explicitly* that he knew of 
neither the experiment nor its outcome when he had already 
convinced himself that the special theory was valid. 

The new method attracted very little attention. On the 
invitation of F. Klein (1849-1925, German), Ricci and his former 
pupil, T. Levi-Civita 1873-1942, Italian), prepared an article on 
the tensor calculus and its applications to mathematical physics 
for publication in a journal read by* mathematicians of all 
nationalities. The article, in French, appeared in 1901. It fell 
rather flat. However, a few curious geometers outside of Italy 
became aware of the new calculus, and at least one, M. Gross- 
mann of Zurich, mastered it and taught it to Einstein. The 
tensor calculus was the particular kind of generalized vector 
algebra appropriate for expressing the differential equations oi 
relativity in covariant form as demanded by a postulate of tne 


The debt of algebra and geometry to general relativity is 2 s 
great as that of relativity to algebra and geometry. Although 
Ricci and Levi-Civita in their expository article of 1901 had 
offered abundant evidence of the utility of tensor analysis in 
applied mathematics, the new calculus was seriously taken up 
by mathematical physicists only after their curiosity had been 
roused by the experimentally verified mathematical predictions 


of relativity. The tensor method quickly induced a vast develop- 
ment of differential geometry. 

Gibbs had predicted in 1886 that vector analysis would some- 
day greatly simplify what in his time was modern higher algebra 
— the theory of algebraic covariants and invariants. He had 
in mind the possibilities of Grassmann’s theory. His prediction 
was verified in the 1930’s. Another prediction of Gibbs of the 
same kind was verified in 1925, when W. Heisenberg found in the 
algebra of matrices the implement he needed for the non-com- 
mutative mathematics of his quantum mechanics. Physicists 
took less kindly to ab ^ ha than they had to tensors; and it 
was a great relief to many when C. Eckart (U.S.A.) and E. 
Schrodinger (Austria) in 1926 showed independently and simul- 
taneously that matrix mechanics could be replaced by wave 
mechanics, in which the theory of boundary-value problems, 
already familiar in classical mathematical physics, is the key 
to the mathematics. 

It scents probable that Grassmann did not anticipate any 
such outcome for his extremely general ‘geometrical algebra.’ 
Two of his successors, Riemann and W. K. Clifford (1845-1879, 
English), both more physical-minded than Grassmann, ventured 
to predict the twentieth-century geometrization of some parts 
of mathematical physics. This was in the middle stage from 
Grassmann to tensors, and it was as remarkable a prophecy as 
any that mathematicians have ever made. But it must not 
be forgotten that mathematicians no less than scientists and 
others have made many false prophecies. The successes are 

Toward structure 

“Mathematics,” according to Gauss in 1831, “is concerned 
only with the enumeration and comparison of relations.” B. 
Peirce (1809-1880, U.S.A.), one of the creators of linear associa- 
tive algebra, asserted 9 in 1S70 that “Mathematics is the science 
which draws necessary conclusions.” Peirce also remarked that 
“all relations arc either qualitative or quantitative,” and that 
the algebra of either kind of relation may be considered inde- 
pendently of the other, or that, in certain algebras, the two may 
be combined. 

These opinions, from what is now a remote past mathe- 
matically, might be admitted by some formalists as anticipations 
of their own conception of mathematics as the theory of struc- 


ture. In particular, the Pythagorean program is superseded. 
Euclid’s postulational method remains. Large tracts of mathe- 
matics have become entirely formal and abstract; the content of 
a mathematical theory is the structure of the system of postu- 
lates from which the theory is developed by the rules of 
mathematical logic, and from which are derived its various 

This excessively abstract view of mathematics evolved from 
the formalization of elementary algebra in the 1830’s, which has 
already been described; the work of Abel and Galois in the theory 
of algebraic equations, of about the same time; the development 
of linear algebra throughout the nineteenth and early twentieth 
centuries; the creation of mathematical logic, beginning with 
Boole in 1847-54 but vigorously pursued only in the twentieth 
century; and finally, from the free invention of non-Euclidean 
geometries after 1825, and the renewed interest in postulational 
methods following Hilbert’s work of 1899 on the foundations of 

Of all these influences, two in particular are germane here: 
the development of linear algebra; and the infiltration of the 
ideas of Abel and Galois into algebra as a whole. The Galois 
theory of equations was acknowledged by both Dedckind and 
Kronecker to be the inspiration for their own general and semi- 
arithmetical approach to algebra. Two of the basic concepts of 
the Galois theory, domains of rationality, or fields, and groups, 
were the point of departure. Both groups and fields will be 
described presently. For the moment we observe the underlying 
methodology -which might have been followed and which would 
be followed today (1945), but which vras not followed histori- 
cally, in the generation of linear algebras, groups, and other 
systems in modern algebra. 

The methodology is that of generalization by suppression of 
certain postulates defining a given system. The system defined 
by the curtailed set of postulates is then developed. Linear 
algebra is obtainable in this way from the algebra of a field. 
Vector algebras, as we have seen, received their initial impulse 
from Hamilton’s suppression of the postulate that multiplication 
is commutative in common algebra. Common algebra is the most 

familiar example of a field. 

Groups also may be derived from common algebra oy tne 
same technique of generalization. But they were not so obtain/— 
originally: and it is doubtful whether they would ever h~ ,c 


attracted the attention they did, had not the momentum of 
history thrust them forward. There are 4,096 (perhaps more) 
possible generalizations of a field. To develop them all without 
some definite object in view would be slightly silly. Only those 
that experience has suggested have been worked out in any 
detail. The rest will keep till they are needed; the apparatus for 
developing them is available. Nevertheless, the postulational 
technique has been one of the most suggestive of twentieth- 
century mathematics; and we shall have occasion to recur to it 
frequently as we proceed. 

Fields being the most familiar of all mathematical systems, 
we shall define them first. A field 14 ( Korper , corpus, corps, do- 
main of rationality) F is a system consisting of a set S of elements 
a, b, c, s, . . . and two operations, ©, O, which ma_v 

be performed upon any two (identical or distinct) elements a, b 
of S, in this order, to produce uniquely determined elements 
a © b and a O b of S, such that the postulates (1) to (5) arc 
satisfied. Elements of S will be called elements of F. For sim- 
plicity, a © b, a G b will be written a -j- b, ab. 

(1) For any a, b of F, a + b and ab are uniquely determined 

dements of F, and b + a — a + b, ba = ab. 

(2) For any a, b, c of F, (a + b) + c = a + (b -f c ), 

( al>)c — a{bc), a(b -f c) = ab -{- ac. 

(3) There exist in F two distinct elements z, v such that 
if a is any element, a + s = a, au — a. 

(4) For any element a of F, there is in F an dement .v such 
that a 4- x = z. 

(5) For any clement a, other than z, of F, there is in F an 
dement y such that ay = v. 

It should be noticed that equality, =, has been assumed as 
a known relation. For completeness: equality is an equivalence 
relation (as defined earlier in connection with congruence). 
That is, if a, b arc any dements of F, a — b or a ^ b, ^ meaning 
'not equal to’; a — a; if a = b, then b — a- if a = b and b — c, 
then a — c. 

This familiar and somewhat elaborate abstraction of common 
algebra and rational arithmetic will serve to illustrate the mean- 
ing of structure and the history of its development. We note 
first that these precise postulates date only from 1903; and that 
in the postulates as given in 1903 (and 1923), the precise meaning 
of cqtialitv is not stated, being taken for granted. In texts of 
1930 or later, it became customary to define equality as an 


equivalence relation before using equality in the postulates of 
a field. This is typical of the continually increasing precision 
in elementary mathematics since the first explicit definition of 
a number field in 1879 by Dedekind. As a final instance of the 
same tendency, it was only in the 1920’s that it became cus- 
tomary to state explicitly that a = b or a ^ b. There is therefore 
little reason to suppose that even these precisely stated postu- 
lates have explicated all the assumptions underlying our habitual 
use of common arithmetic. 

If, in the postulates (1) to (5), the elements, a, b, c, . . be 
interpreted as rational numbers, and u, z as 1, 0, with a + b, ah 
the sum, product of a , b, it is seen that the rational numbers are 
an instance of a field with respect to addition and multiplication. 
Subtraction and division follow from (4), (5). Similarly, ordinary 
complex numbers x + iy furnish another instance; as also do 
Hamilton’s number-couples (#, y) with the appropriate defini- 
tions of u, z, addition, and multiplication, which the reader may 
easily recover. The rational integers 0, ±1, +2, ... do not 
furnish an instance, on account of (5). If F is any field, and 
Xi, . . . , x n are independent variables (or indeterminates), the 
set of all rational functions of Xi, . . . , x n , with coefficients in 
F, is another field. 

With ‘structure’ still not defined formally, it is intuitively 
evident what is meant by the statement that all instances of a 
field have the same structure, and that this structure is as in 
(1) to (5). Further, it is clear that if the logical consequences of 
(1) to (5) are developed, the body of theorems so obtained will be 
valid for each instance of a field. The last is indeed ‘clear,’ 
although a proof of it might be difficult and, as a matter of fact, 
no generally accepted proof had been devised up to 1945. A 
thoroughly satisfactory proof must demonstrate that the rules of 
mathematical logic applied to (1) to (5) will never produce a 
contradiction, such as “a = b and a ^ b.” It seems as if this 
must be the case; but seeming in mathematics is not the same as 
being. ‘Existence,’ for one school, is indeed identified with proof. 

The earliest recognitions of fields, but without explicit defini- 
tion, appear to be in the researches of Abel 15 (1828) and Galois 16 
(1830-1) on the solution of equations by radicals. The first 
formal lectures on the Galois theory were those of Dedekind to 
two students in the early 1850’s. Kronecker also at that time 
began his studies on abelian equations. It appears that the con- 
cept of a field passed into mathematics through the arithmetical 


works of Dedekind and Kronecker. Both, especially Dcdekind, ,<! 
early recognized the fundamental importance of groups for 
algebra and arithmetic. With Dedekind's famous Eleventh 
supplement to the third edition (1S79) of P. G. L. Dirichlet’s 
(1S05-1S59, German) J'orlesungen uber Zahier.tkeorie , the con- 
cept of a number field was firmly established in mathematics. We 
note, however, that Dedekind in this work was interested only in 
algebraic numbers — roots of algebraic equations with rational 
number coefficients. The fields he defined were therefore those of 
real and complex numbers. Kronecker followed in 18S1 with his 
domains of rationality, that is, fields. Although Kroncckcr’s 
definition was more general than Dedekind's, it did not attain 
the complete generality of the postulate system quoted above. 

The passage to final abstractness took about a quarter of a 
century. This need not be traced in detail here; the references 
given are sufficient to orient anyone who wishes to elaborate the 
history. The turning point was Hilbert’s work on the foundations 
of geometry in 1S99. Although this did not concern algebra or 
arithmetic directly, it set a new and high standard of definiteness 
and completeness in the statement of all mathematical defini- 
tions or, what is equivalent, in the construction of postulate 
systems. Compared to what came after 1 900 in this basic kind of 
work, that before 1900 now seems incredibly slack. With abun- 
dant resources at hand to continue the Euclidean program of 
stating explicitly what a mathematical argument is to be about, 
a majority of nineteenth-century mathematicians left their 
readers to guess exactly what was postulated. Neglect to state 
all the intended assumptions incurred its own penalties in faulty 
proofs and false propositions. The change for the better after 
1900 was most marked, but there is still room for improvement, 
especially in mathematics of the intuitive kind — such as the 
repeated appeal thus far to intuition for the meaning of structure. 

Passing to groups, we shall state in full a set of postulates 
for a group, as ‘group’ in the technical sense defined by these 
postulates will occur repeatedly in the sequel. We shall then be 
in a position to define structure. 

A group G is a set 5 of elements a. b, c, . . . , x, y, . . . and 
an operation 0 , which may be performed upon any two (identical 
or distinct) elements n, b of S, in this order, to produce a 
uniquely determined element cOb of S, such that the postulates 
(1) to (5) are satisfied. 

(1) c.0b is in S for every a, b in S. 


( 2 ) aOibOc ) = ( aOb)Oc for every a , b, c in S. 

(3) For every a, b in S there exist x,y'mS such that aOx = b, 
yOa = b. 

These postulates may appear strange to those acquainted 
with others for a group; but they are simpler than some, and all 
are equivalent. Historical notes on groups will be given later; our 
present interest is in mathematics. We proceed to structure, 19 
which seems to have been first recognized, but not defined, in 

Consider two groups, with the respective elements a h b h 
c i, . . . and az, bz, c 2 , . . . and the respective operations 
Oi, 0 2 . These groups are said to be simply isomorphic, or to have 
the same structure, if it is possible to set up a one-one corre- 
spondence between the elements such that, if XiOiyi = Z\, then 
x 2 0 2 yz = Zz, and conversely, where x h y if Zi are the respective 
correspondents of x 2 , yz, z 2 . For further details we must refer to 
the texts. 

This definition is probably the simplest example of what is 
meant by ‘same structure.’ Note that ‘structure’ is not defined, 
but that ‘same structure’ is. For the purposes of algebra this is 
sufficient. If ‘same structure’ seems at first glance to define 
absolute identity, an example to the contrary is supplied by all 
the normal men in a community, all of whom have the same 
shape — two arms, one head, etc. — but no two of whom are 
identical except perhaps topologically. 

A general theory of structure was developed by A. N. White- 
head ( 1861 -, English) and B. Russell 21 (1872 — , English) in 1910 . 
It will suffice here to recall a cardinal definition: A relation P 
between the members of a set x p has the same structure as a 
relation Q between the members of a set y q if there is a one-one 
correspondence between the elements of x v and y q such that, 
whenever two elements of x v are in the relation P to each other, 
their correlates (by the correspondence) in y q are in the relation 
Q to each other, and vice versa. 

If in any division of mathematics there are relations P, 
Q, . . . having the same structure, it suffices to elaborate the 
implications of one, say P, when those of Q, . . . follow on 
translating from P, x p , . . . to Q, y q , . . . by means of the 
relevant correspondence in each case. Each of the postulates of a 
mathematical system can be restated as a relation between the 
data (‘elements’ and ‘operations’) of the system. 

If it is possible to establish a one-one correspondence between 


the postulates of two systems such that correlated postulates 
have the same structure, then the systems are said to have the 
same structure. Instead of saying that two systems have the 
same structure, it is customary in the U.S.A., following E. H. 
Moore (1862-1932, U.S.A.) who used the concept in his lectures 
and writings from about 1893 on, to say that the systems are 
abstractly identical. Abstract identity is itself an equivalence 
relation. If several systems are abstractly identical, obviously 
it is sufficient to develop the mathematics of one in order to 
have that of all. The systems so developed will differ in the 
interpretations assigned to the abstract elements and operations; 
each assignment provides an ‘instance’ of the theory. For exam- 
ple, the algebras of real and of complex numbers, or of Hamil- 
tonian number-couples, are instances of the theory of an abstract 

Tracing the evolution of algebra since the 1830’s, we note a 
constant but largely subconscious striving toward abstractness. 
Concomitantly, abstract identity was sought, sometimes delib- 
erately, as in the theories of groups and fields. Most classification 
is an effort in the same direction preparatory to comparison of 
different theories and the detection of abstract identities. Klein’s 
unification of diverse geometries by the theory of groups in 1872, 
which will be described in connection with invariance, was a 
conspicuous example of the advantages accruing from a recog- 
nized abstract identity. But it seldom happens that anything so 
simple as a group unifies apparently unrelated divisions of 
mathematics with respect to anything deeper than superficialities. 

Enough has been said about structure to indicate what Gauss 
may have had in mind when he observed that “mathematics is 
concerned with the enumeration and comparison of relations.” 
He made this statement in connection with complex numbers. 
On another occasion he expressed a doubt that any ‘numbers’ 
other than the real and complex — such as quaternions, for 
example — would ever be of any use in the higher arithmetic. In 
our further pursuit of algebra and arithmetic we shall be guided 
by this hint of Gauss’, and endeavor to sec what he might have 
had in mind. This, of course, is not the only road by which 
number might be followed from the lS30’s to the twentieth 
century. But following it, we shall have a definite object in view 
by which to orient some of the major trends of algebra and 
arithmetic on the way. 


Arithmetic Generalized 

Continuing with the modern developments of number and 
their influence on the emergence of structure, we shall observe 
next the expansion of modern arithmetic — the Greek arithmetica 
— from its origin in 1831 in the work of Gauss on the law of 
biquadratic reciprocity to its end in mathematical logic. Our 
immediate interest in this chapter is the greatly generalized 
concept of whole number, or integer, which distinguished the 
higher arithmetic of the late nineteenth century from all that 
had preceded it. In a subsequent chapter we shall follow some of 
the main lines of descent of the classical arithmetic from Fermat, 
Euler, Lagrange, and Gauss to the present. Historically, many 
of these older developments preceded the work to be described 
here. But their interest, great though it may be intrinsically, is 
as yet comparatively negligible for mathematics as a whole. 

There are six major episodes to be observed, four of which 
will be described in this chapter and the following. The four are 
the definition by Gauss, E. E. Kummer (1810-1893, German), 
and Dedekind of algebraic integers; the restoration of the funda- 
mental theorem of arithmetic in algebraic number fields by 
Dedekind’s introduction of ideals; the definitive work of Galois 
on the solution of algebraic equations by radicals, and the theory 
of finite groups and the modern theory of fields that followed; the 
partial application of arithmetical concepts to certain linear 
algebras by R. Lipschitz (1831-1903, German), A. Hurwitz 
(1859-1919, Swiss), L. E. Dickson (1874-, U.S.A.), Emmy 
Noether (1882-1935, German), and others. All of these develop- 
ments are closely interrelated. The last marks the farthest 
extension of classical arithmetic up to 1945, and is either the 
climax or the beginning of a structural arithmetization of algebra 




foreseen as early as 1860 by Kroncckcr, but only partly achieved 
by him in the 1880’s. As if in preparation for the climax, the 
algebra of hvpercompicx numbers rapidly outgrew its classifica- 
tory adolescence of the lS70’s, represented by the work of B. 
Peirce and his successors, and became progressively' more con- 
cerned with general methods, reaching a certain maturity' early' 
in the twentieth century'. 

The fifth major episode, which logically' would seem to be a 
necessary' prelude to the others, strangely' enough came last. Not 
until the closing y'cars of the nineteenth century' was any'onc 
greatly perturbed about the natural numbers 1, 2, 3, ... . All 
mathematics, from the classical arithmetic of Fermat, Euler, 
Lagrange, A. M. Legendre (1752-1833, French), Gauss, and 
their numerous imitators, to geometry' and analysis, had ac- 
cepted these speciously simple numbers as ‘given.’ Without 
them, none of the major advances of modern arithmetic would 
ever have happened. Yet no arithmetician asked, “By' whom are 
the natural numbers ‘given’?’’ Kronecker ascribed them to 
God, but this was hardly a mathematical solution. The question 
arose, not in arithmetic, but in analysis. It was answered by the 
modern definition of cardinal and ordinal numbers. This finally 
united arithmetic and analysis at their common source. 

The sixth and last major episode in the evolution of the num- 
ber concept was the application of arithmetic to the differential 
and integral calculus. It is a point of great interest, as will be 
seen in a later chapter, that one of the strongest initial impulses 
for the final application of arithmetic to analysis came from 
mathematical physics. Fourier’s theory of heat conduction 
(1S22) disclosed so many' unforeseen subtleties in the concepts 
of limit and continuity that a thorough overhauling of the basic 
ideas of the calculus was indicated. Many toiled at this for the 
rest of the nineteenth century. It was gradually perceived that 
the cardinals and ordinals 1, 2, 3, . . . demanded clarification. 
By’ 1902 the last subtle obscurity then uncovered had been 
removed, only' to make room for a yet more subtle. The arith- 
metic of 1, 2, 3, ... , and with it mathematical analysis, 
resigned its soul to the searching mercies of mathematical logic. 

About twenty-five centuries of struggle to understand num- 
ber thus ended where it had begun with Pythagoras. The modern 
program is his, but with a difference. Pythagoras trusted 1, 2, 
•’>•*• to ‘explain’ the universe, including mathematics; and 
the spirit animating his ‘explanation’ was strict deductive rca- 


soiling. The natural numbers are still trusted by mathematicians 
and scientists in their technical mathematics and its applications. 
But mathematical reasoning itself, vastly broadened and deep- 
ened in the twentieth century beyond the utmost ever imagined 
by any Greek, supplanted the natural numbers in mathematical 

When, if ever, mathematical logic shall have surmounted its 
obscurities, the natural numbers may be clearly seen for what 
they ‘are.’ But there will always remain the possibility that any 
unsealed range may conceal a higher just beyond; and if the past 
is any guide to the future, arithmeticians will come upon many 
things to keep them busy and incompletely satisfied for the nest 
five thousand years. After that, perhaps, it will not matter to 
anyone that 1, 2, 3, . . . ‘are.’ 

Generalized divisibility 

The class of positive rational integers 1, 2 ,3, . . . was first 
extended, as a class of integers , by the adjunction of zero and the 
negative rational integers —1, —2, —3, .... We recall that 
Euclid in the fourth century b.c. proved one of the cardinal 
theorems concerning positive rational primes: If a prime p 
divides the product of two positive rational integers, p neces- 
sarily divides one of them. A rational prime admits as divisors 
only itself and the units 1, —1. The extension of Euclid’s theorem 
to all the rational integers is immediate and need not be recalled. 
But to emphasize the non-trivial character of the generalizations 
of the rational integers by Gauss, Kummer, Dedekind, and 
others, the preceding definitions must be reformulated so as to 
apply to the generalized ‘integers’ in question. It may be re- 
marked that this simple recasting of the definitions of rational 
arithmetic was one of the three most difficult steps toward the 
desired generalization. The other two were a redefinition of 
arithmetical divisibility, as distinguished from division in alge- 
bra, and the closely related problem of selecting from a given 
class of numbers those which are to be defined as integers. 

First, as to units. With ‘integer’ as yet unspecified, a unit 
In a given set of integers is an integer that divides each integer 
in the set. An integers ‘divides’ an integer ft if there is an integer 
y such that ft — erf. 

Second, as to ‘irreducibles.’ An integer oc is said to be irreduci- 
ble if l a = ftyft with ft, y integers, implies that one of ft, 7 is a 
unit and the other is oc. 



Third, as to primes. An integer a is called a prime if it is 
irreducible, and if further the assertion ‘ a divides fiy' implies 
at least one of the assertions ‘a divides /3,’ ‘a divides 7 .’ 

These definitions accord with those for the rational integers. 
But whereas rational primes and rational irreducibles coincide, the 
like is not true for all of the generalized integers to be described. 

The manufacture of definitions is likely to be a profitless 
pursuit unless there is a definite end in view. The goal here is 
the fundamental theorem of arithmetic: the ‘integers’ defined are 
to be resolvable into powers of distinct ‘primes’ in one way only, 
apart from ‘unit’ factors and permutations of the factors. This 
requirement is too drastic for the ‘arithmetic’ of most linear 
algebras; it is that at which the founders of the theory of alge- 
braic numbers aimed. It was to prove unattainable. 

The means by which the original program was replaced by 
another, which accomplished the essentials of what had been 
sought originally, is one of the finest examples of generalization 
in the history of mathematics. The generalization concerned 
the fundamental concepts of common arithmetic, particularly 
‘integer’ and ‘divisibility . 5 

To be of more than trivial significance, any generalization 
in mathematics must yield on appropriate specialization all the 
instances from which the generalization proceeded, and must 
give in addition more than is contained in all of those special 
instances. The profoundest generalizations appear to be those in 
which the interpretations of all the symbols in the structure 
(postulates) of a given system arc changed. The passage from 
rational integers to algebraic integers was of this kind. 

For example, in the theorem of rational arithmetic, “if a 
divides b, then b docs not divide a unless < 7 , b arc units,” a , b, 
(b -A- (T), and the division-relation are all assigned interpretations 
in the generalization differing from those of rational arithmetic. 
But these interpretations arc such that the statement “if a , 
etc. 5 ’ remains true for the new interpretations. 

Tiie extension of rational arithmetic to an arithmetic of alge- 
braic numbers and, considerably later, to a partial arithmetiza- 
tion of linear algebra, originated in two distinct sources: the 
proof by Gauss in 1828-52, or earlier, of the law of biquadratic 
reciprocity: Rummer’s attempt in the 1810’s to prove Fermat’s 
last theorem. We begin with Gauss. 

If there is a rational integer x such that, when f, q are 
given positive integers, x* — <7 is divisible (without remainder) 


by p, q is called an rAc residue of p. Restated as a Gaussian 
congruence, q is an rAc residue of the given p if and only if 
9r = q mod p is solvable for For simplicity, we describe onlv 
the case in which p. q are positive odd primes. Gauss was particu- 
larly concerned with r. = 4, r. = 3. For r. — 2. Legendre’s lav 
of quadratic reciprocity', which Gauss called “the gem of arith- 
metic,” is 

(p\q')(q\p) = (-i) i{ ™ 5 -» 

■where (p\q) denotes 1 or — 1 according as x- ss p mod q is. or is 
not, solvable for and similarly for (o;P) and x- = g mod p. 
Gauss long sought a reciprocity law for r, = 4 as simple as that 
for r, = 2. He found it only when he passed beyond rational 
integers to complex integers. A Gaussian complex integer is a 
number of the form a -f- hi. where a , b are rational integers. De- 
fining units, primes, and divisibility for his complex integers in 
the straightforward way suggested by analogy with rational 
integers. Gauss proved that the fundamental theorem of arith- 
metic holds for integers a hi. By means of these integers he 
was enabled to state the law of biquadratic (n = 4) reciprocity 
concisely. For n — 3 he found an equally simple theory, based on 
"'integers’ a -f- bp , where p is a root of y- -f y t 1 = 0 and a, b 
are rational integers: but he did not publish his results. 

The history of reciprocity laws for r. > 4 would fill a large 
book. This highly developed subject has been cultivated by 
scores of arithmeticians, and it has had a considerable influence 
on the evolution of modem algebra. But as this specialty, rich 
though it may be intrinsically, is rather to one side of the prin- 
cipal advance, we must leave it here with a remark. 

What is essentially the law of quadratic reciprocity was 
known to L. Euler (1707—1783, Swiss) in 1744—6 but was not 
proved by him. 1 He discussed the law more fully in 1783. Legen- 
dre in 1785 attempted a proof, but slipped in assuming as obvious 
a theorem which is as difficult to prove as the law itself. Gauss 
first published a proof in 1801, and gave six in all. For r. > 2, the 
reciprocity' laws depend upon the algebraic number fields enter- 
ing through binomial equations of degree r,. This brings us to the 
next stage in the development of algebraic numbers. A particular 
algebraic number field of degree n is the set ot all rational func- 
tions of a root of a given irreducible algebraic equation ox degree 

r. with rational integer coefficients. 2 

In his attempt to prove the impossibility 
which x, y. z. p are rational integers, xyz A 

of -f m 

0, and p is a prime 


>2, Kummcr in 1849 resolved x p 4* y p into its p linear factors, 

(x -f y) (x -{- ay) ' ' ' (.v + a J>-1 y), 

v.'here a is an imaginary pth root of 1. This led him to extend the 
theory of Gaussian complex integers to the algebraic number 
field defined by a p ~ i 4* cc p ~ : 4* * * * 4* a -f 1 = 0. With ap- 
propriate definitions of integers, primes, etc., in this field, 
Kummcr persuaded himself for a time that he had proved 
Fermat’s last theorem. But, as P. G. L. Dirichlct (1805-1859, 
German) pointed out to him, he had assumed that the funda- 
mental theorem of arithmetic holds for these integers constructed 
from a. For certain primes p the fundamental theorem is valid 
in the corresponding a-field; for others it is not. The complete 
proof (or a disproof) of Fermat’s theorem was still open. 

Undaunted by this totally unforeseen failure, Kummcr 
invented a new kind of number, which he called ‘ideal’ — not 
to be confused with Dedekind’s ideals. There would be no point 
in describing these here, 3 as they are too far off the main road. 
They apply to the particular number fields considered by 
Kummcr in connection with Fermat’s last theorem. 

Making a completely fresh start in the early 1870’s, J. W. R. 
Dedekind (1 S3 1-1916, German) created a theory of algebraic 
integers applicable to the general case of an algebraic number 
field defined by a root of an irreducible equation 

r.oX n -p 4~ " * ‘ +d a = 0 

of any degree r. with rational integer coefficients a 0 , . . . , a n . 
A root of this equation is called an algebraic number of degree 
r.\ if Co = 1, this number is an algebraic integer; if in addition 
a n = 1 (or —1), the algebraic integer is a unit. Note that any 
rational integer r is an algebraic integer of degree 1, since r is 
the root of x — r = 0. 

All this detail has been recalled to indicate that the generali- 
zation from rational integers and rational units to algebraic 
integers of any degree demanded unusual insight. At first glance 
it seems impossible that a number such as ( — 13 -f- \ — 1 1 5)/2 
should have any of the divisibility properties of a common whole 
number. This specimen, being a root of the irreducible equation 
.v : -f 1 3x -r"l — 0, is in fact an algebraic integer of the second 

Algebraic number holds in which there is unique decomposi- 
tion of algebraic integers into primes arc the exceptions. To 


restore the fundamental theorem of arithmetic to the integers of 
any algebraic number field, Dedekind reexamined divisibility 
for the rational integers. This was the critical step, leading to the 
invention of what Dedekind called ideals. 

An (integral) ideal of an algebraic number field A is a subset, 
say a, of all the integers of F such that, if a, ft are in a, and £ is 
any integer in F, then a — (3 and a£ are in a. The ideal a is said 
to divide the ideal b if every integer in b is also in a, that is, if 
a, considered as a class, contains b. The unit ideal is the set of all 
integers of F; it divides every ideal. An ideal $ is prime if, and 
only if, p and the unit ideal are the only ideals dividing p. 

Unique factorization was restored thus — ‘replaced’ would be 
more strictly accurate. If a is any integer of F, the set of all 
ar£, where £ runs through all integers of F, is easily seen to be an 
ideal. This ideal, denoted by (a), is called the principal ideal 
corresponding to a; and it follows immediately from the defini- 
tions that, if a, (3 are any integers of F, a divides /3 when, and 
only when, (a) divides (/3). ‘Divides,’ in l a divides /3, 5 means that 
there is an integer 7 of F such that (3 = ay; ‘divides,’ in l {a) 
divides (/3),’ means that the principal ideal (a) contains the 
principal ideal (/3); and the theorem asserts that each of these 
division-relations implies the other. 

In rational arithmetic, for example, ‘3 divides 12’ is equiva- 
lent to ‘the class of all integer multiples of 3 contains the class of 
all integer multiples of 12.’ Again, if a, b are given integers, the 
class of all integer multiples of a contains the class of all integer 
multiples of b if and only if a divides b. 

Decomposition of an algebraic integer into a product of 
algebraic integers is now mapped onto a decomposition of an 
ideal into a product of ideals. The fundamental theorem of 
arithmetic is valid in the map. The mapping is as follows. 

The integers d, (3 of F are replaced by their corresponding 
principal ideals (a), ((3), . . . . Since the capital theorem of 
Dedekind’s theory establishes the unique decomposition of any 
ideal (in F) into a product of powers of prime ideals, each of 
(a), Q3), . . . has such a unique decomposition . 4 

Roughly, the crux of the matter is the replacement of the 
relation of arithmetical divisibility by the relation of class- 
inclusion as in either classical or symbolic logic. And, still 
roughly, this replacement is in part responsible for the appear- 
ance of ideals as linear sets of particular kinds in modern algebra 
and in algebraic geometry. 



The invention of ideals has been given what may seem more 
than its legitimate share of space because it is an admirable and 
easily described example of the modern tendency to generaliza- 
tion. The most characteristic detail, possibly, is that of replacing 
the familiar concept of arithmetical divisibility by another that 
includes it. A central relation is replaced by another bearing 
no superficial resemblance to the first. Nevertheless, after the 
replacement, a cardinal theorem (unique decomposition into 
primes) is restored, by mapping or one-one correspondence, to 
a domain in which, before the replacement, the theorem did not 
hold generally. And further, the replacement leaves essentially 
unaltered those cases in which the theorem held before the 

From another point of view, the replacement of the set of 
algebraic integers by the set of correlated principal ideals intro- 
duces uniformity and brings apparent anomalies under a new 
and wider law. An earlier instance of the same procedure oc- 
curred in the introduction of ideal elements (points, lines, 
planes, ... at infinity) into projective geometry during the 
first half of the nineteenth century. Such elements have only a 
remote connection with algebraic number ideals, but in both 
cases the methodology of generalization by extension to regular- 
ize exceptions is the same. 

A feature of this theory that strikes those approaching it for 
the first time as rather peculiar is characteristic of much of 
Dcdckind’s thinking about number: a strictly finite problem is 
solved in terms of infinite classes. The problem for algebraic 
integers is that of unique decomposition; Dedckind’s solution is 
through the particular infinite classes of algebraic integers which 
he called ideals. His theory (1S72) of the real number system is 
based on a similar escape from the finite to the infinite by means 
of what he called cuts. To define "V/I, for example, Dcdckind 
imagined all rational numbers to be separated into two classes, 
say L, U\L contains all those rational numbers, and only those, 
whose squares arc less than 3, and U all those, and only 
those, whose squares arc greater than 3;L, U are said to define a 
‘cut’ in the system of all real numbers, and this particular cut is 

* r~ 

said to define \3. 

The arithmetic of Dcdckind cuts is a map of the usual prop- 
erties of real numbers, such as \2 X "Vi = \2 X 3, familiar 
to analysts and algebraists through centuries of formal manipuia- 


tions. The purely formal work with irrationals produced con- 
sistent numerical approximations, and was sufficient for scientific 
applications of analysis. Dedekind aimed to provide a sound 
logical basis for the traditional formalism of number. The out- 
come of his efforts was a deeper formalism of the infinite. In 1926, 
the leading mathematician of his age, D. Hilbert (1862-1943, 
German) asserted that “The significance of the infinite in mathe- 
matics has not been completely clarified.” Nor had it by 1945. 

Further developments 

Dedekind’s theory of ideals was but one of several con- 
structed for the purpose of restoring the fundamental theorem of 
rational arithmetic to algebraic numbers. The other which has 
survived 5 is Kronecker’s theory of 1881, already mentioned in 
connection with complex numbers. Both Kronecker’s and Dede- 
kind’s theories have extensive ramifications in other departments 
of mathematics, and both exerted a decisive influence on the 
development of modern abstract algebra. 

A third theory has become prominent since its creation in the 
1900’s by K. Hensel, in which numbers are represented by power 
series. This theory originated in the remark that any rational 
integer can be developed into a series of positive integral powers 
of a given prime p , with coefficients chosen from 0, 1, ... , 
p — 1. It may be considered as the ultimate extension of the 
Babylonian, Mayan, and Hindu place-systems of numeration 
in common arithmetic. Analogies with the theory of functions 
of a complex variable, also with the theory of algebraic functions 
of one variable and their representation on Riemann surfaces, 
appear to have guided this arithmetical theory in its rapid 
development. ‘Algebraic function’ is used here in its customary 
technical sense: if P(zv, z) = 0, where P is a polynomial, w is 
called an algebraic function of z. We shall see later that the 
detailed study of such functions and their integrals was a major 
activity of nineteenth-century mathematics. 

It may be of interest to indicate very briefly how the concept 
of arithmetical divisibility as generalized to class-inclusion by 
Dedekind and Kronecker became significant in departments of 
mathematics far distant from arithmetic. Any detailed descrip- 
tion soon becomes highly technical, and we can give only enough 
to suggest that far-reaching applications might have been 
anticipated from the finished form of Dedekind’s theory and the 
broad outline which Kronecker left of his. 


A familiar example from elementary analytic geometry offers 
the plainest hint. If C„(x, y) = 0(n = 1, 2 , . . . , ir.) are the 
equations of m given plane curves, then 

/i (*, y)Ci(x, y) + * * * -bfm(x, y)C a (x, y) = 0, 

in which thc/’s are functions of x, y (or constants), not identi- 
cally zero, is the equation of a curve passing through the points 
common to the m given curves. For simplicity, let all the C’s 
and/’s be polynomials in x, y. Then the system of all polynomials 
/i(.v, y)Ci(x, }') + ••• + /„(*, y)Cn(x, y), in which the C’s are 
held the same and the/’s arc constants or range over all poly- 
nomials in x, y, contains, or ‘divides,’ any particular polynomial 
in the system. ‘Divides’ here is as in Dedekind’s ideals or Kro- 
ncckcr’s modular systems. 

A modular system is a set M of all polynomials in s variables 
.Vi, . . . , x, defined by the property that if P, Pj, P 2 belong to 
the system, then so do Pi *r P; and QP, where Q is any poly- 
nomial in Xi, .... .v,. One further definition enables us to state 
a capital theorem of modern algebra. A basis of a modular system 
M is any set of polynomials Pi, P 2 , ... of M such that every 
polynomial of M is expressible in the form 

R\B\ -{- PjP; -{***"} 

where Pi, P;, . . . are constants or polynomials (not necessarily 
belonging to M). Hilbert’s basis theorem of 1890 states that 
every modular system has a basis consisting of a finite number of 
polynomials or, equivalently, a polynomial ideal has a finite 

Anyone might be excused for doubting this theorem until he 
had followed its remarkably simple proof. In fact, when Hilbert 
applied it to prove the fundamental theorems for algebraic 
forms, P. Gordan (1S37-1912, German), who had previously 
obtained the same theorems by laborious calculations, exclaimed 
“This is not mathematics; it is theology!'’ 

There was a double-edged truth in Gordan’s protest. Hil- 
bert’s theorem marks a major turning point in algebra. It was 
the first example to attract universal attention to the modern 
abstract non-calculating method. Gordan’s proofs were by 
highly ingenious algorithms; Hilbert's attacked the structure 
of the systems concerned — algebraic forms and their covariants 
and invariants. The algorisiic method was incapable of revealing 
the general underlying principle of which Gordan’s theorems are 


but special manifestations. We shall return to this when we 
consider invariance. 

The sharper edge of Gordan’s protest was felt only in the late 
1920’s. A proof in theology, it may be recalled, usually demon- 
strates the existence of some entity without exhibiting the 
entity or providing any method for doing so in a finite number of 
humanly performable operations. Mathematics, particularly 
analysis, abounds in proofs theological in this sense. To Kro- 
necker, all theological proofs in mathematics were anathema. He 
insisted that those existence proofs are invalid and therefore 
worthless in mathematics which do not provide a method for 
exhibiting, or constructing, in a finite number of humanly per- 
formable operations, the mathematical object whose existence is 
alleged to be proved. To most algebraists it is intuitive that a 
polynomial P(x ) with rational coefficients either is or is not 
rationally reducible — the product of two polynomials in x with 
rational coefficients. Kronecker would not admit this statement 
until he had devised a method for deciding in a finite number of 
steps whether a polynomial is actually reducible or irreducible. 

Since Kronecker first demanded constructive existence 
proofs, it has been suspected by some that the free use of ‘theo- 
logical’ existence proofs may lead to inconsistencies. In par- 
ticular, the admissibility of Hilbert’s non-constructive existence 
proof for his basis theorem was questioned in the 1930’s, although 
neither he nor the majority of working mathematicians sensed 
anything objectionable or dangerous in the continued application 
of the theorem. Without it, a vast tract of modern abstract 
algebra and a considerable amount of algebraic geometry would 
evaporate into nothing. A finitely constructive existence proof 
of the basis theorem had not been given up to 1945. The implied 
doubts in this lack are of a piece with those arising from the 
work of the nineteenth century on the real number system. 

None of these deep uncertainties deters mathematicians in 
their technical labors, any more than an occasional eruption 
discourages the vineyardists on the slopes of Etna and Vesuvius. 
The periodic upheavals and submersions under rivers of incan- 
descent lava, are indeed regarded as blessings, except by the 
generations who must endure them. The decomposing lava 
revitalizes the exhausted soil, and the grapes produce a richer 
wine. But it is rather unpleasant for those who must be suf- 
focated or incinerated in order that their successors may prosper. 
Much of the mathematics of the nineteenth and twentieth cen- 



tunes seems now to be significant chiefly because it may 
contribute to a sounder mathematical prosperity in the twenty- 
first century. But we have no assurance that it will. In the 
meantime, our generation endures or enjoys metamathematics 
and continues to create new mathematics. And so has it been 
since existence proofs svere first questioned. 

The general gain to 1910 

What may be called the second heroic age of the theory of 
algebraic numbers ended in the 1870’5-1880’s in the work of 
Dedekind and Kroncckcr. The first great age was that of Gauss 
and Kummer in the 1830’s-l 840’s. The principal innovations in 
each of these periods naturally inspired a considerable number of 
technical developments. But it docs not appear that any funda- 
mentally new concept, or any novel approach comparable in 
general significance to those indicated, was suggested earlier 
than the third great epoch beginning in 1 910 and the I920’s. It 
must be recalled here that we arc interested primarily in the 
development of mathematical thought as a whole, rather than 
in the detailed exploitation of special fields. Before passing on 
to the third period and its relation to the general progress, we 
may glance back at the first two and note once more their origin, 
in order to see their principal residue. Perhaps the most signifi- 
cant contribution is the methodological approach in both 

The theory of algebraic numbers originated in two definite 
problems concerning the rational integers: the laws of n - ic 
reciprocity, designed to yield criteria for the solvability of 
binomial congruences .r” ~ r mod m\ the proof or disproof of 
Fermat’s last theorem. A solution 5 of the first problem for « 
prime was given in what long remained its classic form by 
F. M. G. Eisenstcin (1823-1852, German) in 184*1—50, and by 
Kummer in 1850-61. Thus, in this direction, algebraic numbers 
accomplished the purpose for which they were invented. The 
underlying structure of the modern theory of reciprocity laws, 
dating from about 190S, is that of the modernized Galois theory 
of fields and finite groups. 

The second problem — Fermat's last theorem — responsible 
for the theory of algebraic numbers has resisted the best efforts 
of three generations of arithmeticians since Kummer made the 
first notable progress. In this direction, then, the theory has not 
attained its goal, although it has found much on the way. Of 


both problems it seems fair to say that they, as definite ends, 
have waned in interest, while the methods devised for their 
solution have steadily waxed in importance for modern mathe- 
matics. Algebraists, for example, who have but a slight interest 
in either reciprocity laws or Fermat’s theorem constantly use 
the machinery (fields, ideals, rings, etc.) devised in the first 
instance to handle these problems. 

The like is true of the Galois theory of equations. Galois 
himself made the terminal contribution so far as algebraic equa- 
tions are concerned, and subsequent reworkings of his initial 
theory have added nothing basically new to his criteria for solva- 
bility by radicals. Even the modernized presentation of the 
Galois theory, as in the streamlined model of E. Artin (Ger- 
many, U.S.A.), is a tribute to the mathematical creed of Galois, 
in its elimination of all superfluous machinery. For this modern 
release from algebraic calculation, the direct approach of A. E. 
‘Emmy’ Noether (1882-1935, Germany, U.S.A.) in the 1920’s 
was primarily responsible. Much of her mathematics was in the 
spirit of Galois. But his methods, sharpened and generalized 
by his successors, have transcended the problem for which they 
were invented, and have rejuvenated much of living pure 

It is to be noticed concerning the vital residue of the theory 
of algebraic numbers that it, like the Galois theory, can be 
traced to definite, highly special problems. Neither Galois nor 
the creators of the theory of algebraic number fields set out 
deliberately to revolutionize a mathematical technique; their 
comprehensive methods were invented to solve specific problems. 

Such appears to have been the usual path to abstractness, 
generality, and increased power. Some difficult problem that 
has appeared in the historical development of a particular sub- 
ject is taken as the point of departure without any conscious 
effort to create a comprehensive theory; repeated failures to 
achieve a solution by known procedures force the invention of 
new methods; and finally, the new methods, having been neces- 
sitated by a problem which appeared in the historical develop- 
ment, themselves pass into the main stream. 

Both Fermat’s last theorem and the arithmetical theory of 
reciprocity are but very special cases of a central problem in 
diophantine analysis. It is required to devise criteria to decide 
in a finite number of non-tentative steps whether or not a given 
diophantine equation is solvable. The extreme complexity of 



ihe theories invented for the two special cases suggests that only 
insignificant progress toward a solution of the general problem is 
likely without the invention of radically new methods. 

The contribution from algebraic equations 

The third great epoch in the extension of arithmetic is that 
of the twentieth century after 1910. To anticipate, the introduc- 
tion of general methods into linear algebra, beginning in the first 
decade of the twentieth century, prepared that vast field of 
mathematics, first opened up by Hamilton and Grassmann in the 
1840’s, for partial arithmetization in the second and third 
decades of the century. In 1910, E. Steinitz (1871-?, Germany), 
proceeding from, and partly generalizing, Kronccker’s theory 
(1881) of “algebraic magnitudes,” made a fundamental con- 
tribution to the modern theory of (commutative) fields. His 
work was one of the strongest impulses to the abstract algebra of 
the 1920’s and 1930’s, witli its accompanying generalized arith- 
metic. The outstanding figure in the later phase of this develop- 
ment is usually considered to have been Emmy Noether 7 
(1882-1935, Germany) who, with her numerous pupils, laid down 
the broad foundations of the modern abstract theory of ideals, 
also a great deal more in the domain of modern algebra. The 
application of this work to the ‘integers’ of linear associative 
algebras affords the ultimate extension up to 1945 of common 

One of the main clues threading this intricate maze is the 
Galois theory of fields as it has developed since 1830. The Galois 
theory of equations itself was the concluding episode in about 
three centuries of effort to penetrate the arithmetical nature of 
the roots of algebraic equations. Accordingly we shall consider 
this first. It is of interest in this connection to recall an opinion 
expressed by Hilbert* in 1893 which still retains its force: 

With Gau*% Jacobi and L. Diridilct frequently and forcefully expressed 
their astonishment at the close connection between arithmetical questions and 
certain algebraic problems, in particular with the problem of cyclotomy. The 
baric reason for these connections is now completely disclosed. The theory of 
algebraic numbers and the Galois theory of equations have their common root 
in the theory of algebraic fields. . . . 

After the solution of the general cubic and quartic in the 
sixteenth century, there appears to have been only one con- 
tribution of lasting significance to the algebraic solution of 


equations before the late eighteenth century. E. W. Tschirn- 
hausen® (or Tschirnhaus, 1651-1708, German) in 1683 applied a 

rational substitution — reducible to a polynomial substitution 

to remove certain terms from a given equation. This generalized 
the removal of the second term from cubics and quartics by Car- 
dan, Vieta, and others. About a century later (1786), E. S. 
Bring (1736-1798, Swedish) reduced 10 the general quintic to 
one of its trinomial forms, x 5 + ax + b = 0, by a Tschirnhaus 
transformation with coefficients involving one cube root and 
three square roots, a result of capital importance in the trans- 
cendental solution of the quintic. 

Euler, about 1770, solved the general quartic by a method 
differing from that of Ferrari. This unexpected success led him to 
believe that the general equation is solvable by radicals. As 
remarked in connection with the Greek problem of trisecting an 
angle, it demanded originality of a high order to doubt the 
possibility of a solution by radicals in the general case. Were 
such a solution for the general quintic possible, Euler no doubt 
■would have found it; for he was without a superior on the manip- 
ulative side of algebra. But the quintic called for a different kind 
of mathematics. As Abel pointed out, failure to solve the general 
quintic by radicals might indicate only incapacity on the part of 
the would-be solver; and no number of failures could be of any 
value as an indication whether the problem was solvable. 

A long stride forward was taken by Lagrange 11 in 1770-1. 
Instead of trying to solve the general quintic by ingenious tricks, 
Lagrange critically examined the extant solutions of the equa- 
tions of degrees 2, 3, 4 in a successful attempt to discover why 
the particular devices used by his predecessors had succeeded. 
He found that in each instance the solution is reducible to that 
of an equation of lower degree, whose roots are linear functions 
of the roots of the given equation and roots of unity. Here at last 
was a seemingly universal method. But on applying his reduction 
to the general quintic, Lagrange obtained a sextic. The degree of 
the resolvent equation, instead of being reduced as before, was 
raised. We see now that this was a strong hint of the impossibility 
of a solution by radicals; but Lagrange apparently missed it. He 
had, however, found the germ of the theory of permutation 

grou P s - , , 

In this discovery, Lagrange took the first step toward tne 
general theory of groups, a step of immeasurably greater signifi- 
cance for mathematics as a whole than a complete disposal of the 



theory of algebraic equations. Permutation groups suggested 
abstract finite groups. These in turn suggested infinite discon- 
tinuous groups, and finally the group concept entered analysis 
and geometry with the invention by M. S. Lie (1842-1899, 
Norwegian) of continuous groups in the 1870’s. The reaction 
upon both the discrete and the continuous divisions of mathe- 
matics was far reaching and profound. With invariance, closely 
related to the group concept, the theory of groups in the nine- 
teenth century transformed and unified widely separated tracts 
of mathematics by revealing unsuspected similarities of structure 
in diverse theories. This, however, belongs to the subsequent 
development of Lagrange’s discovery, and will be considered in 
the proper connections. For the moment we are concerned only 
with the application of groups to algebraic equations. 

To recall briefly the nature of what Lagrange found, let 
.Yi, . . . , „r„ denote the roots of the general equation of degree n. 
Then, if a rational function/ of a*i, . . . , x n is left unaltered by 
all those permutations on Xi, . . . , ,v„ that leave unaltered 
another rational function g of .Vi, . . . , a*„, / is a rational func- 
tion of g and the coefficients of the general equation. 

A set of permutations Si, . . . , S r on a given set of letters 
(as Xi, . . . , x n above) form a group in the technical sense 
already defined, when a product, such as SiS„ of two permuta- 
tions S;, Sj, is interpreted as the permutation which results when 
Si is applied first, and Sj is then applied to the new arrangement 
of .Vi, . . . , x„ generated by S,-. For example, if n = 4, and the 
4 letters arc a, b, c, d , the symbol ( abed) means the permutation 
which takes each letter into its immediate successor, a being 
considered the successor of d in this cycle: a into b, b into c, c 
into d, d into a. The permutation {acd) takes a into c, c into d, d 
into a. Hence ( abed) (acd) takes a into b, b into d, d into c, and 
c into a. Thus ( abcd)(acd) = ( abdc ). The identical permutation 
/, or the ‘identity,’ takes each letter into itself or, what is the 
same, leaves each arrangement of the letters unaltered. The set 
of all possible ?:! permutations of x i} . . . , .v„ is called the 
symmetric group on .v t , . . . , a>.. If .Vi, . . . , x n denote the 
roots of an irreducible equation of degree n, the properties of 
the symmetric group on x t , . . . , .v„ arc the clue to necessary 
and sufficient conditions that the equation be solvable by radi- 
cals. It is impossible to go into details here, and wc must refer to 
any modern text on the theory' of equations or higher algebra. 
The fundamental concept for applications of finite groups to 


algebraic equations is that of a solvable group. The meaning of 
this term will be explained presently. 

Lagrange did not explicitly recognize groups. Nevertheless, 
he obtained equivalents for some of the simpler properties of 
permutation groups. For example, one of his results, in modern 
terminology, states that the order of a subgroup of a finite group 
divides the order of the group. Normal (self-conjugate, invariant) 
subgroups, basic in the theory of algebraic equations and in 
that of group structure, were introduced by Galois, who also 
invented the term c group.’ 

Both Abel and Galois were indebted to Lagrange in their own 
profounder work on algebraic equations. Before Abel set himself 
(1824) the problem of proving the impossibility of solving by 
radicals the general equation of degree greater than four, an 
Italian physician, P. Ruffini (1765-1822), beginning in 1799, had 
attempted to do the same. Ruffini’s definitive effort (1813) is 
said by some who have examined it to be essentially the same as 
Wantzel’s simplification of Abel’s proof. Abel published this 
proof at his own expense in 1824; it was reprinted in 1826 by 
A. L. Crelle (1780-1855, German) in the initial volume of his 
great journal. Remediable defects are said by some competent 
algebraists to mar the final proofs of both Ruffini and Abel. But 
as the oversights are not fatal, it is customary to say that each 
of these two proved the impossibility of solving by radicals the 
general equation of degree greater than four. Their work was 
entirely independent. The unique importance of Abel’s proof is 
that it inspired Galois to seek a deeper source of solvability, 
which he found in the theorem that an algebraic equation is 
solvable by radicals if and only if its group, for the field of its 
coefficients, is solvable. 

We cannot enter into the technicalities of Galois’ theorem. 
But assuming some acquaintance with the modern theory of 
algebraic equations, which, after all, is well over a century old 
as this is written, we shall use a few of its concepts to illustrate 
the meaning of structure as exemplified in this capital theorem 
of algebra. The simple isomorphism of any two groups was 
defined in connection with the postulates for a group. Galois 
considered simply isomorphic groups as the same group which, 
abstractly, they are. A. Cayley 12 (1821-1895, English) in 1878 
expressed this by saying that the properties of a group are 
defined by its multiplication table. 

A subgroup Hi of a group G is said to be a normal divisor of 


2 35 

G if for every s in G, sH\ = II \S, where slh denotes the set of all 
products sh, h in II 1 , and similarly for II\S and the set of products 
hs\ equality here means that the two sets contain the same ele- 
ments. A subgroup of G other than G itself is called a proper 
subgroup. A maximal normal divisor of G is a proper normal 
divisor of G that is not a proper subgroup of any proper normal 
divisor of G. The maximal normal divisors of the group of n! 
permutations of the roots of the general equation of degree n 
appear in the criteria for solvability by radicals. To state the con- 
nection, we require the definition of quotient (or factor) groups. 

The order of a group is the number of distinct elements in 
the group; and Lagrange proved (1770-1) in effect that the order 
of a subgroup divides the order of the whole group. If Hi is a 
normal divisor of order m x of a group G of order n, then must 
n = Wi?ij ?i an integer; and it can be shown that 

G = Ih + silli -{-•••+ s q ^ilh, 

where no two of the sets II i, s x Hi, . . . , have an element 

in common, and the plus signs mean that all the elements of G 
arc separated out into these q x mutually exclusive sets. That is, 
the + is logical addition, as in the Boolean algebra of classes. 
Let A',, A' ; , . . . , A',, denote these q x sets (in any order). Then, 
if KiKj denotes the set of all products formed by multiplying 
an clement of K { by an element of A'/, it can be shown that pre- 
cisely mi of these products are distinct, and that these m } are 
all the elements of some one of the AT’s. Moreover, with multi- 
plication KiKj as just described, K x , A'-, . . . , A 5j form a group, 
called the quotient (or factor) group of G with respect to the 
maximal normal divisor Ih of G. This quotient group is denoted 
by G///i; its order is q x , and the order v.{= m x qi) of G, divided 
by the order (in.) of Ih, is called the index of Ih under G. Thus 
the index of Ih under G is here a j. 

Now there may be more than one proper maximal normal 
divisor of G. If there is, its quotient group can be formed as 
above, and its index under G is known. Our concern here is 
with all of these possibilities at each stage of the process next 

Proceeding with Ih as we did with G, we find its quotient 
group Ih! Ih with respect to any maximal proper normal di- 
visor Ih of //j. This divisor may be only the identity-group con- 
sisting of the single element I (the identity) of G. The process is 
now repeated with Ih. and so on, until it stops automatically 


with I. In this way, starting with G and ending with I, we get the 
sequence of groups G, Hi, H 2 , ... , H t ( = I), each of which 
(after G) is a proper maximal normal divisor of its immediate 
predecessor. There is also determined the sequence of quotient 
groups G/ Hi, HijHi, H 2 /H 2 , . . . , Hi_i/H t , and the corre- 
sponding indices, say qi, q 2 , ... , q t . Two final definitions, 
and we can state several striking consequences of this iterated 
process. A group having no normal divisors except itself and the 
identity-group I, formed of the single element I (the identity of 
the group), is said to be simple. If all the indices q lf q 2 , ... , 
q t are prime numbers, the group G is said to be solvable. 

Remembering that there may be several ways of proceeding 
at each step, we state the following conclusions. First, 13 in what- 
ever way we proceed, we get the same number of groups G, Hi, 
H 2 , . . . Hi. Second, all the factor groups displayed above arc 
simple. Third, 14 in whatever way we proceed, the factor groups 
are the same, although not necessarily in the same order, and 
hence similarly for the indices qi, q 2 , , q t . In a sense which 

need not be elaborated, these theorems of C. Jordan (1838-1922, 
French) (1870) and O. Holder (1859-1937, German) (1889) are 
a remarkable revelation of the structure of any finite discrete 
group. Since 1930 they have been refined and extended in what 
may be likened to the minute anatomy of any articulated organ- 
ism. 15 Recalling the capital theorem of Galois for the solvability 
of an algebraic equation by radicals, we see that these theorems 
go to the root of the matter. To anticipate slightly, it may be 
noted here that the Jordan-Holder theorems have themselves 
been structurally analyzed as phenomena of the theory of 
‘lattices’ or ‘structures.’ This theory will be discussed in the 
next chapter. 

The further development of the theory of groups will be de- 
scribed presently. For the moment we note that the question of 
solvability by radicals received a conclusive answer in the theory 
of finite groups.\ 

After the impossibility of solving the general equation of 
degree higher thian the fourth by radicals had been proved, the 
next problem wa.s to find what kind of functions would suffice to 
solve the general ouintic. The general cubic had long been known 
to be solvable by circular (trigonometric) functions. The circular 
functions are uniform (single-valued) singly periodic functions 
of one variable. They are degenerate forms of the elliptic func- 
tions, which are uniform doubly periodic functions of one 



variable, or ‘argument’ x . If }{x) is an elliptic function, and p\, 
pi arc its two periods, pi/p! is necessarily imaginary, and 
jf(x + «j/ji + n:pi) = f(x) for all choices of the integers «j, n ?. 
As will be seen when we consider analysis, Abel and Jacobi in 
the 1820’s discovered the elliptic functions through the inversion 
of elliptic integrals. 16 An extensive department of the theory of 
elliptic functions is the problem of the division of the periods: 
if there is an integer n such that nx is a period, the problem of 
division by n is to find elliptic functions having x as argument. 
This problem leads to certain algebraic equations which, for 
« = 2, 3, 4, 3.2*, are solvable by radicals. The degree of the 
equation for division by any odd n is Or — l)/2. Thus for n — 5 
the degree is 12. But if n is prime, the equation is obtainable in 
a much simpler form, being only of degree n -f* 1. We recall that 
Lagrange was led to a resolvent equation of degree 6 in his 
attempt to solve the general equation of degree 5 by radicals. The 
problem of division of elliptic functions for n — 5 incidentally 
provided the functions of a , b which reduce the trinomial form 
,v s -f- ox + b — 0 of the general quintic to an identity. The trans- 
cendental functions necessary to solve the general equation of 
the fifth degree had therefore been constructed. 

This unexpected result was found by C. Hermitc (1 822—3 90S, 
French) in 1858. Ilermite was led to it by his intimate knowledge 
of elliptic functions. lie observed that an equation occurring in 
the problem of quinquisection of elliptic functions could be 
transformed into Bring’s form of the general quintic. Simulta- 
neously, Kroncckcr was nearing the name goal by another road. 
Kroncckcr's method differed profoundly from Hermite’s. It was 
closer to what Galois might have done, had he lived. In 1853 
Kroncckcr net himself the task of solving a fundamental prob- 
lem encountered by Abel in his attack on algebraic equations: 
to find the most general function of x } , . . , , x n that can be a 
root of an algebraic equation with coefficient in a given field. lie 
proved that the equations arising from the theory of division of 
certain transcendents suffice to solve the general equations of 
certain degrees, and in this way obtained transcendental solu- 
tions of the general cubic and quartic. He then attacked the 
general quintic without previous reduction of the equation by a 
Tschirnhaus transformation to remove certain terms. His object 
was to find a method capable of extension to equations of any 

Intercut in such problems flagged during the second half of 


the nineteenth century. So far as the general quintic is concerned 
F. Klein (1849-1925, German) in 1884 reviewed 16 all the labors 
of his predecessors, and unified them with respect to the group 
of rotations of a regular icosahedron about its axes of sym- 
metry. The earliest discussion from the standpoint of groups of 
the (modular) equations arising in the division of elliptic func- 
tions was by Galois. 

As a specimen of later results in this same general field, a 
theorem of Hilbert may be cited: the general equation of degree 
nine requires for its solution functions of four arguments. 

In summary, the chief contribution of the theory of alge- 
braic equations to the number concept appears to have been the 
characterization of the irrationalities required for explicit general 
solutions. The attempt to solve the general algebraic equations 
of degrees higher than the fourth in terms of functions con- 
structed from the given coefficients by a finite number of 
additions, subtractions, multiplications, divisions, and root ex- 
tractions ended in the proof by Abel and Ruffini that such 
solutions do not exist. But such solutions — as for the general 
equations of degrees 2, 3, 4 — define certain species of irrationali- 
ties. It therefore became necessary to seek totally different 
kinds of irrationalities to effect the solution of equations of 
degree higher than the fourth. These were found for degree 5 
in the elliptic modular functions. 

The work of Galois and his successors showed that the nature, 
or explicit definition, of the roots of an algebraic equation is 
reflected in the structure of the group of the equation for the 
field of its coefficients. This group can be determined non- 
tentatively in a finite number of steps, although, as Galois 
himself emphasized, his theory is not intended to be a practical 
method for solving equations. But, as stated by Hilbert, the 
Galois theory and the theory of algebraic numbers have their 
common root in that of algebraic fields. The last was initiated 
by Galois, developed by Dedekind and Kronecker in the mid- 
nineteenth century, refined and extended in the late nineteenth 
century by Hilbert and others, and finally, in the twentieth 
century, given a new direction by the work of Steinitz in 1910, 
and in that of E. Noether and her school since 1920. 

Between these later developments and the work of the nine- 
teenth century on algebraic numbers and algebraic number 
fields, the elaboration of hypercomplex number systems inter- 
vened. This will be our concern in the following chapter. 



It is to be noted that the principal clues to the final general- 
ization of arithmetic have been Dedekind’s theory of alge- 
braic number fields, Kroneckcr’s parallel theory of what he 
called “algebraic magnitudes,” and the theories of general fields, 
groups, and rings. A ring differs from a field in that an inverse 
for multiplication is not postulated. The rational integers, 0, 
±1, ±2, . . . are the simplest instance of a ring; the class of 
all rational integers is a group with respect to addition and is 
closed under multiplication. In a general ring, multiplication is 
not assumed to be commutative. Linear associative algebras 
are instances of rings. 

Changing outlooks , 1870-1920 

Allusions to groups will appear frequently as v/c proceed. 
From 1870 to the 1920’s, groups dominated an extensive sector 
of mathematical thought, and were occasionally rather rashly 
touted as the long-sought master key to all mathematics. The 
first date, 1870, marks the publication of C. Jordan’s (1838-1922, 
French) classic Traits des substitutions cl des equations algebriques, 
which contained a great deal more than its modest title indicates. 
This is emphasized because Jordan was one of the few leading 
specialists in groups who made signal contributions to other 
departments of mathematics, including analysis. Others in the 
same category were Klein, Lie, Poincare, G. Frobenius (1849— 
1917, German), W. Burnside (1852-1927, English), and L. E. 
Dickson (1 874 — , U.S.A.). But the great majority of those who 
labored in groups were specialists in the narrowest sense; and 
some of them after 1920 were content to parrot the informed 
but obsolete opinions of well-rounded mathematicians on the 
place of groups in mathematics as a whole, without acquiring 
the knowledge necessary to enable them to form a reasonable 
personal judgment. 

During the fifty years from 1S70 to 1920, mathematics did 
not stagnate; and although groups remained one of the seemingly 
permanent additions to mathematical thought, informed opinion 
after 3920 was less immoderate than it had been in the 1890’s 
in its claims for the domination of groups over all mathematics, 
it was therefore inexcusably misleading to retain in the 1930’s 
inflated estimates of groups that may have been valid in the 
1910’s, as if mathematics had remained stationary since the 
death (1912) of Poincare, even when chapter and verse for 


the estimates in question were cited in the works of some of 
the greatest mathematicians of a bygone generation. 

As a specific instance, a prominent specialist in finite groups 
reproduced in 1935 the long-since-superseded dictum of Poin- 
care that “The theory of groups is, as it were, the whole [our 
italics] of Mathematics stripped of its matter and reduced to 
‘pure form,’” as if this extravagance were the considered verdict 
of competent opinion in 1935. That it was not, will be seen in 
detail when we follow certain developments of geometry since 
1916, to be described in connection with that other outstanding 
addition of the nineteenth century to all mathematical thought, 
the concept of invariance. Poincare’s dictum was a gross exag- 
geration even when it was first uttered. At best, it was an under- 
standable overstatement, possibly for emphasis, that deceived 
nobody above mathematical illiteracy. While we may continue 
to remember the achievements of the great creators in mathe- 
matics with gratitude, we do the masters of the past a left-hand 
honor when we perpetuate their outmoded opinions for the 
misguidance of oncoming generations. 

Some of the more conspicuous landmarks in the development 
of groups after Galois may be noted here, exclusive of the con- 
tinuous groups discussed in a later chapter. Even before Galois 
coined the term ‘group,’ A. L. Cauchy (1789-1857, French) 
made (1815) extensive investigations in what are now called 
permutation groups, and discovered some of the simpler basic 
theorems. He returned to the subject in 1844-6, and just missed 
the fundamental theorem (1872) of L. Sylow (1832-1918, 
Norwegian), proved in most texts on the theory of finite groups. 
Cayley (1854) stated the earliest set of postulates for a group, 
thereby defining groups in the accepted technical sense. This 
definition sank out of sight and, as will be seen in connection 
with continuous groups, some of the leading experts, including 
Lie and Klein, occasionally used the term ‘group’ for systems 
which are not groups in the technical sense now universal. Con- 
sequently, the statements of certain theorems in older work 
require amendment. Another set of postulates was given (1882) 
by H. Weber (1842-1913, German), whose Algebra (3 vols., ed. 
2, 1898-9) presented a masterly synopsis of algebra as it was at 
the close of the nineteenth century. 

In passing, there is no more instructive demonstration of the 
change in outlook and objectives that distinguished the algebra 
of the early twentieth century from that of the late nineteent 



than a comparison of Weber’s classic with an advanced treatise 
of the 1930's. The transition from the old to the new began in 
1910 with the work of Steinitz. It used to be said before 1910 
that a thorough mastery of three famous classics, Weber’s 
Algebra , J. G. Darboux’ (1842-1917, French) Lcqons sur la 
theorie generals des surfaces el les applications geomelriques du 
calcul infinitesimal (2 vols., 1887-8; ed. 2, 1913-15), and E. 
Picard’s (1856-1941, French), Traite d'analyse (3 vols., 1891-6; 
cd. 3, 1922-7), would suffice for a liberal education in mathe- 
matics, and enable a competent student to begin creation in 
what were then topics of living interest to research workers in 
mathematics. Less than a third of a century sufficed to render 
this particular liberal education hopelessly antiquated for any- 
one seeking to orient himself quickly for a creative career in 
vital mathematics. Those who in 1940 would arrive at the front 
of progress were, for the most part, taking short cuts that did 
not exist before 1910, or even before 1920. 

The first decade of the twentieth century witnessed a some- 
what feverish activity in the postulational analysis of groups, 
in which American algebraists produced numerous sets of postu- 
lates for groups, with full discussions of complete independence. 
By 1910, nobody could possibly misunderstand what a group is. 

In another department of finite groups, also, American alge- 
braists were incontinently prolific: the determination of all 
finite groups of a given order, especially all permutation groups 
on a small number of letters. One of the earliest attempts at a 
complete census was that (1858) of T. P. Kirkman (1806-1895), 
an English clergyman in a muggy parish, who claimed that his 
methods sufficed for an exhaustive enumeration. Kirkman will 
appear again in connection with topology. Not a very well- 
known mathematician, although rather a notorious one in his 
own day for the perfection of his sarcasm, Kirkman appears in 
retrospect to have been one of the born combinatorialists in 
mathematical history. For various reasons, he received practi- 
cally no encouragement and about as much recognition. Of 
Americans who made notable contributions to what may be 
called Kirkman’s program, F. N. Cole (1861-1927) and G. A. 
Miller (1S63-) were among the most prolific. 

Groups of linear homogeneous substitutions on n variables, 
also groups of such substitutions with integer coefficients, in- 
cluding congruence groups, were studied by many after Jordan 
(1870) had shown their importance in several departments of 


algebra and analysis, including hyperelliptic functions and the 
geometry of plane quartics. After Jordan’s own work in this 
division, that of E. H. Moore 17 (1862-1932), Dickson, and H. F. 
Blichfeldt (1874—), all of the U.S.A., from the late lS90’s to the 
second decade of the twentieth century, accomplished most. 
Interest in this specialty collapsed after about 1918, and the 
more imaginative algebraists turned their efforts in other 

By 1920, the painstaking collection and detailed analysis of 
special groups had become a thing of the past. Should the fruits 
of all this devoted labor of about half a century ever be required 
in either pure or applied mathematics, the toilers of the future 
will be spared decades of some of the hardest and most thankless 
drudgery ever successfully carried through in the history of 
algebraic computation. Finite groups are an episode in modern 
combinatorial analysis and, as such, are as difficult to civilize 
as any other phenomena in that inchoate science. 

A brilliant exception to purely combinatorial methods ap- 
peared in 1896-9 in the algorithm of group-characters invented 
by Frobenius, and applied by him and others 18 with conspicuous 
success to several difficult problems in finite groups. The neces- 
sary computations, although often tedious, are non-tentative 
and non-combinatorial. They may therefore presage a more 
intelligently reasoned, less grubbing attack on the problem of 
group structure than that of the taxonomic period. With the 
appearance of groups in the early 1930’s in quantum mechanics, 
the somewhat neglected algorithm of Frobenius became of 
possible scientific significance, and the heavy labor of applying 
it in detail to the permutation groups required in physics was 
undertaken. So possibly science may stimulate the algebra of 
the future to devise more practicable methods of calculation 
and enumeration in finite groups than those of the heroic age 
of uninspired hard labor. 

Mathematics and society 

In reviewing the contribution of algebraic equations to the 
development of the number system, any mathematician today 
must be impressed by the apparent permanence of the ideas 
introduced by Abel (1803-1829) and Galois (1 SI 1—1 832), and 
the profound difference between their approach to mathematic^ 
and that of their predecessors including, in some respects, Gauss 



(1777-1855). To these young men, perhaps more than to any 
other two mathematicians, can be traced the pursuit of general- 
ity whch distinguishes the mathematics of the recent period, 
beginning with Gauss in 1801, from that of the middle period. 
They initiated for the whole of mathematics the deliberate 
search for inclusive methods and comprehensive theories. Their 
forerunners in the middle period were Descartes with his general 
method in geometry; Newton and Leibniz with the differential 
and integral calculus created to attack the mathematics of 
continuity by a uniform procedure; and Lagrange, with his 
universal method in mechanics. Their contemporary in recent 
mathematics was Gauss, who in his arithmetic sought to unify 
much of the uncorrelated work of the leading arithmeticians from 
Fermat to Euler, Lagrange, and Legendre. Both Abel and Galois 
acknowledged their indebtedness to the theory of cyclotomy 
created by Gauss; and although they went far beyond him in 
their own algebra (Abel in analysis also), it is at least conceivable 
that neither Abel nor Galois would have chosen the road he 
followed had it not been for the hints in the Gaussian theory of 
binomial equations. 

Both Abel and Galois died long before their time, Abel at the 
age of twenty-seven from tuberculosis induced by poverty, 
Galois at twenty-one of a pistol shot received in a meaningless 
duel. When Abel’s genius was recognized, he was subsidized by 
friends and the Norwegian government. By nature he was genial 
and optimistic. Galois spent a considerable part of his five or six 
productive years in a hopeless fight against the stupidities and 
malicious jealousy of teachers and the smug indifference of 
academicians. Not at first quarrelsome or perverse, he became 

Whoever, if anybody, was responsible for the colossal waste 
represented by these two premature deaths, it seems probable 
that mathematics was needlessly deprived of the natural success- 
sors of Gauss. \\ hat Abel and Galois might have accomplished 
in a normal lifetime cannot be even conjectured. That it would 
have been much and of the highest quality seems probable. Early 
maturity and sustained productivity are the rule, not the excep- 
tion, for the greatest mathematicians. It may be true that 
the most original ideas come early; but it takes time to work 
them out. Gauss spent about fifty years developing the inspira- 
tions that came to him (this is substantially his own descrip- 
tion) before he was twenty-one, and even with half a century of 


Emergence of Structural 

With the material already described as a background, we 
shall now observe in some detail the trend toward ever greater 
generality and more refined abstraction which distinguished 
much mathematics of the recent period from nearly all that 
preceded 1S-10. Structure, in a sense to be noted and described, 
was the final outcome of this accelerated progression from the 
particular to the general. The entire movement may be seen in 
geometry as clearly as in algebra and arithmetic, and will be 
remarked in that connection in later chapters. The course fol- 
lowed here continues that already indicated merely for con- 
venience. It is typical of that for all divisions. 

Wc saw that Gauss in 1851 invented his complex integers 
.Vj + :.%•» to solve a specific problem in rational arithmetic. His 
.v» -f t.V; written as a number-couple (xi, x : ) suggested hyper- 
complex numbers (.\*j, . . . . ay) with n coordinates -Vj, . . . ,.r n ; 
and it was natural to ask whether any of these extended num- 
bers, with real or ordinary complex coordinates, might be useful 
in rational arithmetic. More generally, how may ‘integers’ be 
defined in a system of hypcrcomplcx numbers, and what is their 

Before cither problem could be attacked, or even formulated 
precisely, the algebra of hypcrcomplcx number systems had to be 
developed. But this in its turn was not a definite problem. Once 
the algebraic problem had been made precise, its solution 
followed rapidly. 

1 here appear to have been three principal phases after the 
problem was first posed by the invention of quaternions. 


Three phases in linear algebra 

The first phase was represented by such work as that of 
B. Peirce in 1870, which sought to find and exhibit all the linear 
associative algebras in a given (finite) number of fundamental 
units. 1 

The second phase, merging into the third, began in the first 
decade of the twentieth century and continued to about 1920. In 
this period, general theorems applicable to all linear associative 
algebras were the objective. 

The third phase was distinguished for its restatement in 
abstract form of much that was already known, and the intro- 
duction of arithmetical concepts, such as ideals and valuations, 
into the resulting abstract algebra. The outcome 2 was an exten- 
sive and intricate theory assignable to either algebra or arith- 
metic, according to taste. The algebraic number rings and fields 
long familiar in the theories of algebraic equations and algebraic 
numbers; the ideals and dual groups of Dedekind; the relative 
fields of Hilbert; the modular systems of Kronecker; and the 
Galois theory of fields, all contributed to the final abstract 
theory. The finished product exhibits the broad outlines of the 
theories from which it evolved as but different, particularized 
aspects of a unified whole, like the varying projections of an 
intricate geometrical configuration on a moving plane. In addi- 
tion, the abstract theory gives a wealth of results not obtainable 
from its classical instances. 

The abstract method 

The entire development required about a century. Its 
progress is typical of the evolution of any major mathematical 
discipline of the recent period; first the discovery of isolated 
phenomena; then the recognition of certain features common to 
all; next the search for further instances, their detailed calcu- 
lation and classification; then the emergence of general principles 
making further calculations, unless needed for some definite 
application, superfluous; and last, the formulation of postulates 
crystallizing in abstract form the structure of the system 
investigated. The detailed elaboration of the abstract system 
implicit in the postulates then proceeds undistracted by what 
may be adventitious circumstances in any special instance. 
Incidentally, this is the reason that our extremely practical 
decadic Hindu-Arabic numerals are a positive detriment, except 


for numerical checks, in investigating the properties of numbers. 
The p-adic and g-adic numbers of Hensel are closer to arithmetic. 

The full import of the abstract formulation appears only 
when it is taken as the point of departure for the deliberate 
creation of netv mathematics. Certain postulates in the original 
set are suppressed or contradicted, and the consequences of the 
modified set are then worked out as were those of the original. 
For example, in a field as first defined, multiplication is commuta- 
tive. This raises the question whether it is possible to construct 
consistent ‘algebras’ subject to all the postulates of common 
algebra except the commutativity of multiplication. Again, 
in common algebra (a commutative field), if a 0, and ab = ac, 
it follows by division that b — c. But division by a presupposes 
that a has an inverse with respect to multiplication. Division is 
not defined in a ring; nevertheless there arc rings in which, if 

e ~ 0 

and ab — ac , then b = c. Hence the existence of inverses as in a 
field is not postulated, but is replaced by the weaker condition 
just stated taken as a postulate. The result is a type of algebra 
more general than a field, in that it is based on some but not all 
of the field postulates or their consequences. 

While viewing the abstract method we need not let our 
sincere admiration for its undeniable beauties betray us to the 
fate of Narcissus. It is just possible that our descendants may 
record that we perished of hunger while staring at the seductive 
reflection of our own superficial ideas — 

“What thou scest, 

What there thou scest, fair creature, is thyself — ”, 3 
or that we fell in and were drowned in something we never sus- 
pected just below the entrancing surface. To illuminate this 
heresy with an even more heretical example, what reason is there 
for supposing that because Dedekind's ideals did what was 
required for algebraic number rings, something very much like 
them should be introduced into other rings? Is it even evident 
that the arithmetic of a non-commutative ring should be based 
on ideals at all ? Or that the usual definition of integral elements 
of a ring by close analogy with the like — by means of the rank 
equation — for an algebraic number ring is the most promising 
clue: The obvious retort is to demand something better of the 
doubter. Admitting the justice of this, we may nevertheless 
consider the other possibility. 


The root of these doubts seems to be the unimaginative lack 
of a clearly recognized objective. If the aim is merely to create 
new theories which many find intensely interesting and even 
beautiful, then the abstract method keeps on reaching its goal. 
In this respect it somewhat resembles Stephen Leacock’s hero 
who leapt on his horse and dashed furiously off in all directions. 
But the taste of another generation may find our abstractions 
boring and our beauties vapid. They will have a hard way to 
travel before coming upon something different. The tangled 
masses of our theories will impede their every step. If they are to 
progress, their only possible road will circumvent our work al- 
most entirely. The like happened once before, when Descartes 
walked clear round the synthetic labors of Euclid and Apollonius. 
To the skeptically inclined, viewing the vast accumulations in 
abstract geometry, abstract algebra, and abstract analysis of the 
twentieth century, another Descartes seems about due. Unless 
he arrives within the next two thousand years, no two mathe- 
maticians in the world twenty centuries hence will understand 
each other’s words. In the meantime we may appreciate the 
tremendous accomplishments of the modern abstract method 
and trace the main steps by which it finally arrived since Hilbert 
gave it the strongest impulse since Euclid in his geometry of 

The first phase in the development of abstract algebra, that 
of calculation and tabulation, is on the same scientific level as 
systematic botany. All sciences of the past seem to have been 
condemned to creep through this Linnaean stage of development. 
If only Linnaeus would stay as dead in mathematics as he 
appears to be in biology, mathematics might be far leaner and 
more virile than it is. But the irrepressible botanist keeps on 
rising from the dead; it is impossible to keep him down. Idle the 
vanguard of mathematical progress is advancing to genera, a 
host of straggling camp followers busies itself with the collection 
and classification of trivial or discarded subspecies. It rvas so in 
the theories of algebraic in variants and finite groups. An exas- 
perating instance from recent analysis is the introduction of tv.o 
new technical terms to distinguish a: > 0, x S 0. After enough 
classification has been done to indicate some inclusive character- 
istic, there would seem to be no point in collecting further 
specimens unless they are to be used. Linear algebra fortunate!) 
escaped the intensest fury of the taxonomists in its rapid passage 
to the second phase. 


Totvard structure in algebra 

We shall select a few typical episodes from the history of 
linear algebra* to illustrate the general trend since about 1870, 
in nearly all mathematics, from the detailed elaboration of 
special theories to the investigation of interrelations between the 
theories themselves. Technicalities are unavoidable, "but they 
may be ignored by those unfamiliar with the subject; the 
important items are not the facts but the relations between 
them. Even without technical knowledge, it is possible to appre- 
ciate distinctions in scope and generality between different 
theorems. The technicalities, however, are of interest on their 
own account. Some of them are landmarks in their own province. 
Only what appear to have been the principal steps leading from 
the enumerative stage of hypercomplex number systems to the 
abstract theory of all such ‘algebras’ will be considered here. 

The work of B. Peirce (1S70) aimed at principles for the ex- 
haustive tabulation of linear associative algebras in a given finite 
number of fundamental units, with real or complex number 
coefficients. His methods were adequate, and his partial failure 
to find all the algebras he sought in less than seven units was due 
to mere oversights. 

Peirce’s problem was equivalent to that of exhibiting all sets 
of n linearly independent symbols (the basal, or fundamental, 
units) €\ forming closed systems under associative 
multiplication, on the assumption that any product e T e, is a 
linear function of Ci, , e n . For n given, the problem can be 
solved brutally by actually constructing all possible multipli- 
cation tables, applying the associativity condition 

— {e T e,)e t , 

and retaining only those which actually dose. The labor of such 
an undertaking quickly becomes prohibitive with increasing n, 
and Peirce proceeded otherwise. Two of his guiding principles 
depended on the presence or absence of nilpotent and idempotent 
units: if there is a positive integer r > 1 such that c T — 0, Peirce 
called c nilpotent; and similarly for e- — c and idempotent. 
These pervasive concepts of linear algebra were the foundation 
of Peirce’s classification. His work was continued bv his son, 
C. S. Peirce (1859-1914, U.S.A.), who also made outstanding 
contributions to mathematical logic and is said to have invented 
the peculiarly Yankee philosophy known as pragmatism. 


In an appendix to his father’s memoir, C. S. Peirce proved 
a famous theorem, 5 which we restate in its customary form: The 
only linear associative algebras in which the coordinates are 
real numbers, and in which a product vanishes only if one factor 
is zero, are the field of real numbers, the field of ordinary com- 
plex numbers, and the algebra of quaternions with real coeffi- 
cients. The advance here beyond tabulation is evident. 

The theorem also indicates one possible kind of answer to the 
question asked by Gauss. It suggests that quaternions might 
have a useful arithmetic, as they are so closely related to ordinary 
complex numbers. Extensive arithmetics of ordinary quaternions 
were constructed by R. Lipschitz (1832-1903, German) in 1886, 
and by A. Hurwitz in 1896. Dickson in 1922 simplified these 
arithmetics. The historical order here is that of increasing 

From the late 1870’s through the 1890’s, linear algebra took 
several new directions which looked extremely promising at the 
time, but which do not appear to have influenced the main 
advance significantly. Thus G. Frobenius (1849-1917, German) 
in 1877 developed an interesting connection between hypercom- 
plex number systems and bilinear forms. This was followed in 
1884 by H. Poincare’s discovery of a similar connection with the 
continuous groups due to M. S. Lie, (1842-1899, Norwegian). 
Lie’s outstanding contributions to nineteenth-century mathe- 
matics will be noted in connection with invariance. Poincare 
replaced the problem of classification by an approachable 
equivalent for a wide class of linear algebras: to find all con- 
tinuous groups of linear substitutions whose coefficients are 
linear functions of n arbitrary parameters. This line of attack 
was developed with considerable succesjs by G. W. Scheffers 
(1866-, German), a scientific legatee of Die, who in 1891 under- 
took to show that Lie’s theory of (finite) continuous groups 
contains the theory of hypercomplex / number systems. The 
theory of such groups therefore afforded principles for the 
classification of linear associative algebras. 

In the 1890’s Lie’s theory was perhaps more assiduously cul- 
tivated than it was to be for a generation, when it was revived in 
a renovated differential geometry. Reflecting this widespread 
interest, E. Cartan (1869-, French) in. 1898 applied the Lie 
theory to obtain a classification of hypercomplex number sys- 
tems which followed ‘naturally’ from that theory. But sue 
apparently promising leads were practically abandoned short v 


after 1900, and the main advance proceeded in another direc- 
tion, beginning in 1907 with the work of J. H. M. Wedderburn 
(1882-, Scotch, U.S.A.). 

Frobenius and Cartan had obtained numerous special results 
for hypercomplex number systems in which the coordinates are 
rational numbers, whereas in a general theory the coordinates 
should be elements of nothing more restricted than an abstract 
field. The postulates for such a situation were formulated in 1905 
by Dickson. As much of Cartan’s development depended on the 
characteristic equation of a linear associative algebra, it was not 
always extensible to the general case. In any event, algebra took 
a new turn in 1907, heading directly toward a theory of structure. 

The theorem of Frobenius and Peirce on quaternions sug- 
gested the search for all linear associative® algebras satisfying 
certain preassigned conditions. The most important of these are 
the division algebras, in which, if a(y£ 0) and b are any elements 
of the algebra, each of the equations ax = b, ya = b has a unique 
solution. Two of the earlier results on division algebras may be 
recalled, to provide concrete examples of structure. 

Galois initiated the study of fields containing only a finite 
number of distinct elements. It was proved in 1893 by E. Ii. 
Moore (1862-1932, U.S.A.) that every finite commutative field 
is of the type considered by Galois; that such a field is uniquely 
determined by a pair of positive integers p, n, of which p is 
prime; and that the corresponding field contains p n distinct 
elements. This theorem exhibits one of the many variants of 
structure: an exhaustive characterization is prescribed for all 
(commutative) fields containing only a finite number of distinct 
elements. Another variant appears in Wcdderburn’s theorem of 
1905, that if the coordinates of a linear associative division 
algebra arc elements of a finite field, then necessarily multiplica- 
tion in the algebra is commutative. 

As will be seen presently, division algebras play a dominant 
part in the theory of algebraic structure. The determination of 
all division algebras, or the discovery of comprehensive classes 
of such algebras, thus became a central problem in the theory. 
Dickson in 191-1 constructed division algebras in r. fundamental 
units with coefficients in any field F. 

At this point we may repeat that we arc primarily interested 
here in the emergence of the theory of structure, particularly as 
exemplified in linear algebra. To bring out the essentials, it will 
be necessary next to state a rather formidable-looking theorem 


containing several technical terms whose meaning has not been 
explained. Those already acquainted with the subject will rec- 
ognize the statement as one of the fundamental theorems' 
(Wedderburn’s, 1907) on the structure of algebras. Those seeing 
it for the first time may substitute letters S, X, Y, . . . for 
the technical terms ‘sum/ ‘semi-simple/ ‘direct sum/ . . . , as 
we shall do in a moment, and attend solely to the construction 
of the sentences constituting the theorem. The structural charac- 
ter, which is our concern here, is then evident. As a mere verbal 
convenience, a linear associative algebra whose coordinates are 
in a field F is said to be over F. The theorem states that: 

(1) Any linear associative algebra over a field F is the sum 
of a semi-simple algebra and a nilpotent invariant subalgebra, 
each over F ; 

(2) A semi-simple algebra over F is either simple or the 
direct sum of simple algebras over F. 

(3) Any simple algebra over F is the direct product of a di- 
vision algebra and a simple matric algebra, each over F, includ- 
ing the possibility that the modulus is the only unit of one 
factor. 8 

Eliminating the technicalities, irrelevant for our purpose, 
we restate this with S, X. Y, . . . . 

(1) Any linear associative algebra over a field F is the S of 
an X-algebra and a E-algebra, each over F; 

(2) An X’-algebra over F is either a Z-algebra or the DS of 
Z-algebras over F: 

(3) Any Z-algebra over F is the DP of a W- algebra and a 
17-algebra, each over F. 

The theorem exhibits the structure of any linear associative 
algebra over any (commutative) field F, in terms of three kinds 
of operations, S, DS, DP, and five species of algebras X, Y, Z, 
TV, U. Thus any linear associative algebra over any field F may 
be dissected into algebras of the five kinds specified, and always 
in the same way, namely, by S, DS, DP. Without further 
elaboration, this is what is meant by saying that all linear 
associative algebras over any field F have the same structure 
with respect to certain specified kinds of subalgebras. Attention 
may therefore be confined to the five specified kinds of sub- 
algebras. One of these, W, comprises the division algebras. 

The radical distinction between general structure theorems of 
this kind and the cataloguing type of algebra which preceded it is 
obvious. The shift of objective is typical of modem abstract 


mathematics. Specimens are no longer prized for their own 
curious sake as they were in the nineteenth century. It is as if 
some industrious company of fossil collectors who had never 
heard of Darwin were suddenly enlightened by an evolutionist. 
Their interesting but somewhat meaningless collections would 
simplify themselves in an unsuspected coherence. 

Toward abstraction in analysis and geometry 

Three further advances of 1906-20 toward abstractness and 
generality may be mentioned here, as they are connected with 
the real numbers. None originated in algebra; yet at least one — 
J. Kiirschak’s (1864—1933, Hungary) — was to suggest far-reach- 
ing consequences in both algebra and arithmetic. All were in the 
direction of a theory of valuation generalizing that of real 
numbers and ordinary complex numbers. 

With any ordinary complex number x + iy, written as a 
Hamiltonian number-couple (x, y), is associated the unique real 
number |(.x, y)|, the ‘absolute value’ of ( x , y), which is the posi- 
tive square root of x~ + y~. But if x, y arc elements of an ab- 
stract field F, x : -f- y 2 is not a real number. To distinguish the 
zero clement of F from the zero, 0, of the real number field, we 
shall write it O’. 

Extending the properties of absolute values for the field of 
ordinary complex numbers to the elements O', x, y, . . . of 
an abstract field, F, Kiirscbak in 1913 associated with any 
clement z of F a unique real number, its ‘absolute value,’ 
denoted by jd, subject to the postulates [O'j = 0; ].vj > 0 if x ^ 0‘; 
i-v.j = Jcj \tt\i\z + tej S H + pi, for any z, tv in F. 

The last is sometimes called the triangular inequality. It is 
the analogue of the theorem in plane Euclidean geometry that 
any side of a triangle is less than, or equal to (when the vertices 
are collincar), the sum of the other two sides. If the ‘distance’ be- 
tween any two elements X, y be defined by |.v — yj, the postulates 
for these absolute values reproduce the properties usually 
associated with the concept of distance. They were so defined in 
1906 by (R.) M. Frcchct (1878-, French) in his thesis for the 
doctorate at Paris. This work is one of the sources of modern gen- 
eral or abstract analysis, the theory of abstract spaces, and to- 
pology (all to be noted in other connections). A further advance 
in this direction was made by S. Banach (1892-1941, Polish) 
in 1920, who removed the restriction that the elements x, y, . . . 
be in a field; his .v, y, . . . are elements of any class whatever. 


It might be thought that nothing not already known could 
issue from such faithful copies of the simplest properties of real 
and complex numbers. To substantiate the contrary, we need 
cite only one rather unexpected outcome of this process of ab- 
straction. It has been found that much of the analysis based on 
the real or complex numbers has its image in general analysis. 
It is not necessary to assume that x, y, . . . are real or complex 
numbers to obtain many results which formerly were supposed 
to be consequences of that assumption. In Frechet’s analysis, 
for example, limit points and convergent series are definable, 
and the theorems on convergence for this abstract analysis are 
applicable to the special case of the analysis in which the basic 
elements are real or complex numbers. Again, it might be 
imagined that for the rational numbers r, . . . the only possible 
frjj is the familiar \r\. But it was shown (1918) by O. Ostrowski 
(1893-, Russian) that there are in fact precisely two distinct 
types. Thus, in this instance at least, abstraction led to some- 
thing new and unexpected. 

A terminus in arithmetic 

Gauss’ question concerning the possible utility of hyper- 
complex numbers in the higher arithmetic is of peculiar interest, 
both historically and mathematically. We have seen that Gauss 
was acquainted with quaternions, and we have referred to 
applications of quaternions to the classical theory of numbers. 
It is unlikely that Gauss had developed quaternions far enough 
to suspect that they have interesting arithmetical properties. 
Nevertheless, he asked 9 “whether the relations between things, 
which furnish a manifold of more than two dimensions, may not 
also furnish permissible kinds of magnitudes [numbers] in 
general arithmetic?” 

There has been much speculation as to what Gauss con- 
sidered 'permissible.’ Each guess generated its own answer in an 
enumeration of all algebras satisfying the ‘permissible’ condi- 
tion. We shall report only one of several closely similar answers, 
as the direction taken by general arithmetic since the time of 
Gauss certainly was not foreseen by him. 

Weierstrass is said to have proved the following theorem in 
his lectures of 1863; at any rate he published it, with several 
more of a like kind, in 1884. The only hypercomplex number 
systems with real coordinates, in which a product vanishes only 
if at least one of its factors does, and in which multiplication is 


commutative, are the algebra with one fundamental unit c such 
that e- = e, and the two-unit system of ordinary complex num- 
bers. If Gauss did not permit divisors of zero, and if he insisted 
on commutative multiplication, this answered his question. The 
algebra with e- — c is of little interest; the other gave Gauss his 
own arithmetic of complex integers a -f- hi, a , b real integers. In 
short, he himself had reached the end of the road he may have 
imagined but did not explicitly indicate. 

Further progress was in other directions, starting in the 
work of Dedekind on algebraic number fields and ideals, and in 
Kronccker’s theory of modular systems as developed by himself 
and others, notably E. Lasker (1S6S-1941, German, former 
world chess champion) in 1905, J. IConig (1849-1913, Hungary) 
in 1903, and F. S. Macaulay (1862-1937, English) in 1916. We 
pass on to a short enumeration of what appear to have been 
the principal steps toward these vast developments of modem 
algebra and arithmetic. 

Newer directions 

From the great mass of work that has been done since 1900 
on the arithmetization of algebra — or vice versa — we shall 
select only three items, to indicate the trend toward abstractness 
and the analysis of structure. 

In his algebraic theory of fields (1910), E. Stcinitz sought 
all possible types of fields and the relations between them. Pro- 
ceeding from the simplest types, explicitly defined, he extended 
these by cither algebraic or transcendental adjunctions. In al- 
gebraic adjunctions, Stcinitz followed the method of Cauchy as 
exploited by Ivroneckcr, described in an earlier chapter. The con- 
cepts of characteristic, prime field, complete field, and others 
now familiar in modern texts on higher algebra were fully treated 
in this profound reworking and extension of Kronecker’s theory 
of algebraic magnitudes. The final outcome may be roughly de- 
scribed as an analysis of the structure of fields with respect to 
their possible subficlds and supcrfields. 

The next item, dating from about 1920, marks a distinct ad- 
vance. It is represented by a host of vigorous workers who, in 
the twentieth century, undertook to do for an abstract ring what 
Dedekind had done for any ring of algebraic numbers, and to ex- 
tend the Galois theory to abstract fields. Thus the Dedckind 
theory of ideals was abstracted and generalized, as was also the 
Galois theory. The first of these may properly be assigned to 


arithmetic, as one of the chief objectives is the discovery, for any 
ring, of unique decomposition theorems analogous to the funda- 
mental theorem of arithmetic, or to the unique representation 
of a Dedekind ideal as a product of prime ideals. It was not to be 
expected that there would be a single type of decomposition 
obviously preferable above all others for a general ring; nor was 
it reasonable to suppose that rational arithmetic or the theory of 
algebraic numbers would be translatable into the new domain 
with only minor modifications. Only the arithmetic of rings with 
commutative multiplication had been discussed with anything 
approaching completeness up to 1945. 

In spite of radical differences between the arithmetic of 
commutative rings (usually without divisors of zero, the so- 
called domains of integrity) and that of algebraic numbers, the 
theory of Dedekind ideals proved a valuable clue. For example, 
in the Dedekind theory, an ideal has a finite basis; that is, any 
number of the ideal is representable as nibi -f • • * -f n,b r) 
where hi, . . . , b r are fixed integers of the algebraic number 
field concerned, and n-i, fir range independently over all 

integers of the field. But this fact does not directly suggest a 
profitable generalization to rings. However, it is equivalent to 
the theorem that any sequence A\, Ai, A 3 , . . . of ideals which 
is such that is a proper divisor of Aj, for/ = 1, 2, ... , 
ends after a finite number of terms. This ‘chain theorem,’ valid 
in Dedekind’s theory, is generalizable. 

Two basic but rather inconspicuous-looking items of the 
classical theory of algebraic number ideals passed unchanged into 
the abstract theory, the G.C.D. (‘greatest’ common divisor) and 
L.C.M. (‘least’ common multiple). Although at the first glance 
these are mere details, experience has shown that they are the 
framework of much algebraic structure and that, when their 
simplest properties are restated abstractly as postulates, the 
resulting system unifies widely separated and apparently dis- 
tinct theories of algebra and arithmetic. They lead, in fact, to 
what seemed the most promising theory of algebraic-arithmetic 
structure devised up to 1945. We shall therefore describe their 
properties in some detail. 

The relevant phenomena appeared first in mathematical 
logic, specifically in the algebra of classes, now called Boolean 
algebra after its founder, G. Boole (1815-1864, English). If the 
letters A, B, C, . . . denote any whatever classes, and if the 
symbol > is read ‘includes,’ or ‘contains,’ it is true that from 


A > B and B > C follows A > C. The symbol < is read ‘is 
included in,’ or ‘is contained in.’ 

If A, B are any two classes, their ‘intersection,’ denoted 
by [A, 2?j, is the most inclusive — largest — class whose members 
arc in both A and B. For example, if A is the class of all animals 
with red hair on their heads, and B is the class of all girls, [A, B] 
is the class of all red-headed girls. If A is as before and B is the 
class of all vegetables, [A, B] is the null class — the class with no 
members. Again, if A , B are any classes, their ‘union,’ denoted 
by ( A , B), is the least inclusive class whose members are in A, 
or in B, or in both. In the second of the above examples, (A, B) 
is the smallest collection each of whose members is either an 
animal with red hair on its head, or a vegetable. 

We may restate these definitions as follows. If A, B are any 
two classes, they have a unique most inclusive common subclass, 
\A , B], and a unique least inclusive common superclass, {A, B). 
By definition, A is equal to B , -written A — B, if, and only if, 
A > B and B > A-, and by convention, A > A. It is now a 
simple exercise in language to verify the following statements 
concerning any set (or class) <3 of classes A , 2?, C, D, D\, . . . , 

M, Mi, . . . 

(1) If A, B, C are any three members of ©, such thaty/ > B 
and B > C, then A > C. 

(2) If A, B are any members of ©, there is a member of <3, 
say D, such that D g A, D £ B\ and such that, if also g. A, 
Di g B, then D x g D. There is also a member of ©, say M, such 
that M S A, M § B ; and such that, if also Mj ^ A, Mi Si B, 
then M\ § M. 

The assertions in (2) arc true when D is the intersection 
\A, 2?] of A , B, and 21/ is their union ( A , B). Reading ( A , ( B , C )) 
as the union of A and ( B , C), and [A, [2?, C]] as the intersection 
of A and [B, C], we have the following theorems as immediate 
consequences of (1), (2): 

[A, B), {A, B) are uniquely defined; [A, B] — [B, A], 
(A, B) = (2?, A)- [A, A] — A, {A, A) « A; [A, [2?, CJ] = 
\[A, 2?], C], (A, (2?, C)) = ((v/, 2?), C); ( A , [//, 2?]) = A, 
[A, (A, B)] = A. 

It may be left to the reader’s ingenuity to decide whether or 
not the next is true for classes, 

(3) If A < C < (A, B ), then C = (A, \B, q). 

It is important for our purpose to verify that (1), (2), and the 
simple theorems above arc satisfied for the following wholly 


different interpretations of A, B, C, . . . , >,=,<; A, B 
C, . . . is any class of positive rational integers; ‘ = ’ is equality 
as in common arithmetic; ‘ >’ means ‘divides’; ‘ <’ means ‘is 
divisible by’; [A, B] is theL.C.M. of A , B, and ( A , B) their G.C.D. 
Recalling that ‘divides’ in the theory of Dedekind ideals means 
‘contains’ or ‘includes,’ as in inclusion for classes, we see why 
the interpretation in terms of classes should be relevant for the 
theory of ideals. 

We now empty A, B, C, <, [A, B], ( A , B) of all 

interpretation and of all meaning, and take (1), (2) as postulates 
defining the meaningless marks A, ...,>, <, etc. Denoting 
the D , M in (2) by [A, B], and (A, B) respectively — a mere 
convenience of notation — we can deduce from our postulates 
the same theorems as before. 

The abstract system so defined is not vacuous; for we have 
actually exhibited two of its instances. One would have sufficed. 
There are many more. The abstract system defined by (1), (2) 
has been called by various names, including ‘structure’ and 
‘lattice.’ The second is to be preferred here, to avoid confusion 
with ‘structure’ as previously defined in mathematical logic. 

Here we reach the ultimate in arithmetized algebra or alge- 
braized arithmetic up to 1945. It may turn out to be a very bad 
guess but, as this is written, many of the younger generation of 
algebraists and arithmeticians believe that in this theory of 
lattices they have at last unified a welter of theories inherited 
from their prolific predecessors. The theory has been most 
vigorously developed in the United States. To vary the historical 
monotony of dwelling almost exclusively on the dead, we may 
mention the names of two of the most active contributors to this 
rapidly expanding domain of abstract mathematics, G. Birk- 
hoff 10 and O. Ore. 

The assertion (3), which was abandoned to the ingenuity of 
the reader, is not a consequence of (1), (2) in the abstract theory. 
When (3) is included with (1), (2) among the postulates, the 
resulting system defines a special type of lattice which is named 
after Dedekind, because he was the first (1897) to investigate 
such systems. That he did so, is but another instance of his 
penetrating and prophetic genius. His work passed practically 
unnoticed for a third of a century, when its significance was 

realized in the theory of lattices. _ . , 

To leave this comprehensive theory of lattices here with no 
indication of its scope would do it but scant justice. A i ery ne 


quotation 11 from one of the creators of the theory contains the 
meat of the matter. “In the discussion of the structure of alge- 
braic domains, one is not primarily interested in the elements of 
these domains, but in the relation of certain distinguished sub- 
domains, like invariant [normal] subgroups in groups, 15 ideals 
in rings, and characteristic moduli in modular systems. For all 
of these are defined the two operations of union and intersection, 
satisfying the ordinary axioms” — the postulates (1), (2). 

The rapid expansion of this theory of structures or lattices, 
after a quiescence of about a third of a century following 
Dedckind’s introduction of dual groups, is typical of much in 
the recent development of mathematics. Among other common 
features is the apparently inevitable slowness with which a 
basic simplicity underlying a multitude of diversities finally 
emerges. Usually the unifying concept implicit in all of its 
different manifestations is almost disconcertingly obtdous once 
it is perceived. But there is as yet no recognized technique for 
perceiving the obvious or for not confusing significance with 
triviality, and each instance, it seems, must wait its own more 
or less random occasion. Nor are mathematicians always reliable 
prophets of what mathematics is to retain its vital interest or 
acquire a new importance. An example which deserves to become 
a historical classic is the sudden rise to popularity of the tensor 
calculus which, until the relativists adopted it, was the neglected 
waif of the mathematicians. In all such revaluations it is easy 
after the fact to sec that they might have occurred much earlier 
than they did. Yet nobody can foresee where the next is to 
happen. Mere neglect, however, is not necessarily an assurance 
of immortality to any who may be overanxious about their 
reputations or the permanence of their work. 

In the matter of lattices or structures, one of the deter- 
mining characteristics of the theory might have been anti- 
cipated — but was not — in 1854 when Boole published his Laws 
of thought. The irrelevance of the nature of the individual ele- 
ments of the classes whose unions and intersections give Boolean 
algebra its distinctive character was recognized by Boole himself. 
Apparently without fully realizing what he had done, and what 
now seems so plain, Boole had taken the first and the decisive 
step toward the abstract algebra and some of the geometry of 
the 1930’s-1940's. In these newer developments the primary 
interest is not in the elements of certain domains, but in the 
inclusions, the intersections, and the unions of distinguished 


subdomains formed of classes of elements of the original domain. 
Sets of elements rather than the elements themselves become 
the basic data, and the original level of abstraction rises to the 
next above it in what experience has shown to be a natural 
hierarchy of abstractions. 

The Boolean algebra of classes attracted but little attention 
from mathematicians during the nineteenth century, and the 
hints it might have offered projective geometers and algebraists 
passed unnoticed. If Boole had assimilated the controversy of 
the 1820’s over the validity of the geometric principle of duality, 
he might well have anticipated the algebraic interpretation of 
projective geometry of the 1930’s. The controversy will be 
noted in a later chapter; for the moment it suffices to state that 
the ‘real space’ some of the contestants imagined they were 
discussing evaporated. We shall return to this presently. 

Another clue that might have been noticed appeared in the 
various decomposition theorems of arithmetic and algebra. 
Decomposition, as in unique factorization in rational arithmetic 
and the theory of algebraic integers, reduces a given system 
to a system of simpler parts, the parts being combined according 
to prescribed rules to produce the elements of the given system. 
The ‘simpler parts’ in rational arithmetic are the rational primes; 
in algebraic numbers,, the prime ideals; and in both, the rules 
are those of multiplication as in an abelian group. The decom- 
position may not be unique, as for instance the basis of an abelian 
group. Further suggestive examples of decomposition were the 
Jordan-Holder theorem, described earlier, for finite groups, 
Wedderburn’s theorems on the structure of linear algebras, 
and A. E. Noether’s determination of all commutative rings 
in which there is unique prime ideal factorization. For the 
Jordan-Holder theorem it is the subsystem of normal (invariant, 
self-conjugate) subgroups of a given finite group with respect to 
which the decomposition is effected that matter; the elements 
of the group itself are of only minor importance. For algebraic 
integers the distinguished subsets are the ideals, and so on; 
in each instance the original elements are subsidiary. With these 
and other examples before them, it gradually became evident to 
algebraists that some common characteristic must be the ulti- 
mate source of at least a part of the phenomena of decomposition 
in the several theories. Axiomatic formulations reveal the com- 
mon characteristic to be an underlying lattice. In his work of 
1900 on the dual groups generated by three moduli, Dedekind 


had noticed the affiliation with what is now called Boolean alge- 
bra. Once the unifying feature was recognized, the natural next 
step was to develop the algebra of lattices as an independent 
theory on its own merits. The outcome was an abstract theory of 
structures or lattices. 

Abstraction for its own sake may prove fruitful or barren. If 
it also suggests new theorems or expresses what is already known 
more simply and more clearly, abstraction rises to the level of 
creative mathematics. As the theory of lattices advanced, it 
unified and simplified much of existing algebra and the funda- 
mental parts of certain other disciplines; it also aided in the 
discovery of new results. It was not to be expected that the 
general lattice by itself would clarify all the phenomena of de- 
composition, and as the theory progressed, specialized lattices 
of various types were defined to accommodate particular theories. 
One significant clue was the invariance of the chain length — 
the number of factors of composition — in the Jordan-Holder 
theorem for finite groups. Other equally patent clues might 
have been followed initially. But all this is in retrospect, and 
it will be in closer conformity with time if we merely mention 
some of the items which the theory of lattices illuminated. 
For further details we may refer to G. Birkhoff, Lattice theory, 

Boolean algebra, the historical source of lattice theory, 
found its natural place in the theory as a special type of lattice 
(complemented, distributive). Distributive lattices may be 
realized by rings of sets, an observation — in other terminologies 
— which goes back substantially to Euler. From this follows, 
on appropriate specialization, the representation theory of 
Boolean algebras by fields of sets. Partly suggested by the last, 
there arc applications of lattice algebra to the classic theories 
of sets and measure. In a later chapter we shall note the wide 
generalizations of the geometric ‘space’ of the nineteenth cen- 
tury which evolved into the twentieth-century geometry' of 
abstract space. Among these generalizations, the function 
spaces of the twentieth century' readily accommodated them- 
selves to abstraction. L. Kantorovich (Russian) essentially' 
defined (1957) a partially ordered linear space as a (real) linear 
space having non-negative elements /, symbolized / > 0, subject 
to the following three postulates: If / > 0 and X > 0, then 
V > 0; if / > 0 and — / > 0, then / = 0; if / > 0 and g > 0, 
then / -r £ > 0. It was shown by G. Birkhoff in 1940 that 


every function space then known forms a lattice with respect 
to this partial ordering. Vector lattices were also defined and 
shown to be decomposable into their suitably defined positive 
and negative components. In connection with abstractions of 
the absolute values of real and complex numbers, we shall note 
in a later chapter the type of abstract space named for the 
Polish mathematician S. Banach. Banach lattices were defined 
as vector lattices with a suitably specialized norm (generalized 
absolute value), and it was shown that all the examples given 
by Banach of his space are such lattices. As a last result in this 
direction, the decompositions of partially ordered function 
spaces when characterized abstractly yield components forming 
a Boolean algebra. 

From its historical origin, lattice theory might have been 
expected to have applications to mathematical logic and the 
mathematical theory of probability, and such was found to be 
the fact. A curious application (1936) provided a model of 
quantum mechanics. We recall that the concept of observables 
is central in this mechanics, also that the indeterminacy prin- 
ciple imports probability into the metaphysics of all physical 
observation. A basically different application (1944) of logic 
and probability to the quantum theory will be noted in the 
concluding chapter. 

A revealing application of lattices was the restatement of 
projective and affine geometries by K. Menger in 1928, Bemer- 
kungen zu Grundlagenfragen IV, as instances of what was later 
to be called lattice algebra. Menger considered a system of 
abstract elements for which two associative, commutative 
operations, denoted by +, are defined. The operations admit 
neutral elements, the ‘vacuum’ V and the ‘universe’ U, such 
that A V = A — A • U for all A in the system, and it is 
postulated that A A = A = A • A. The characteristic fea- 
ture of the algebra is the postulate of ‘absorption’: if A + B 
= B , then A ■ B = A, and conversely for all A, B in the system. 
Thus the algebra is essentially what G. Birkhoff (1934) called 
a lattice. Birkhoff also (1934) independently reduced projective 
geometry to a topic in lattice algebra, and Menger (193o) 
published a somewhat amplified account of his theory of 1928, 
in which it had been stated that the algebra is applicable to the 
theories of measure and probability. So far as the reduction of 
projective geometry to lattice algebra is concerned, this now 
seems an inevitable but unanticipated climax of the thorough- 


going axiomatization of projective geometry — not very well 
known in Europe, apparently — by 0. Veblen and J. W. Young 
in 1910. 

The first two decades of the twentieth century witnessed an 
unprecedented activity in axiomatics, especially in the U.S.A., 
consequent on the work of D. Hilbert (1862-1943, German) 
in the foundations of geometry. The first volume of the Projective 
geometry (1910) at Veblen and Young subjected projective 
geometry to a logical rigor it had not experienced before, recon- 
structing this department of geometry as a hypothetico-dcduc- 
tive abstract system in accordance with Hilbert’s general 
formalistic program for all mathematics. Although geometric 
intuition may not have been abandoned in the rigorous treat- 
ment, it was not recognized, either officially or unofficially. 
The opening sentences of the work assert that 

Geometry deals with the properties of figures in space. Every' such figure is 
made up of various elements (points, lines, curves, planes, surfaces, etc.), and 
these elements bear certain relations to each other (a point lies on a line, a line 
passes through a point, two planes intersect, etc.). The propositions stating 
these properties are logically interdependent, and it is the object of geometry' 
to discover such propositions and to exhibit their logical interdependence. 

A more inclusive but somewhat vaguer description of geom- 
etry by one of the authors cited will be reported in a later 
chapter. The above passage is sufficient for the present; it 
almost begs to be translated into the language of lattices. 
In the expressive slang of 1945, it is a natural for such transla- 
tion by anyone who has ever glanced at the postulates for a 
lattice. Yet essentially these postulates were accessible ten years 
before the above manifesto of the aims of geometry was printed, 
and it was not until a quarter of a century after it appeared 
that the connection between lattices, or Dedckind’s dual groups, 
and geometry was perceived. But again this overlooking of 
what is now plain almost to the verge of truism is no reflection 
on the perspicacity of creative geometers. Rather is it merely 
another instance of the historical commonplace, emphasized 
long ago by W. Bolyai in connection with the final emergence 
of non-Euclidean geometry after centuries of seemingly unneces- 
sary struggle, that mathematical discoveries, like the spring- 
time violets in the woods, have their season which no human 
effort can retard or hasten. 

A further quotation from the Projective geometry of 1910 
disposes, in the current mathematical manner, of certain debates 


on the nature of ‘space’ that have exercised metaphysicans for 
many centuries: “Since any defined element or relation must be 
defined in terms of other elements and relations, it is necessary 
that one or more of the relations between them remain entirely 
undefined; otherwise a vicious circle is unavoidable.” Although, 
as will be seen in connection with ‘space’ of any finite number 
of dimensions, it is not necessary to choose ‘points’ as the 
ultimate undefined elements of geometry, they usually were 
chosen and frequently still are. Lines, etc., then became classes 
of points. Geometry, even of the elementary-school kind, was 
never primarily concerned with the elements in these classes, 
but with the intersections and unions of the classes. The lines, 
surfaces, etc., were the distinguished subdomains of the initially 
structureless chaos of points with which geometry actually 
dealt. However, points were usually in the background, if not 
in the mind, of the geometer doing geometry — implicitly 
Veblen’s definitions of ‘geometer’ and ‘geometry’ — ever since 
the time of Euclid. According to Euclid, “ a point is that which 
has no parts and has no magnitude.” But this nihilistic attempt 
at a definition was not incorporated into the deductive develop- 
ment of geometry until, as noted by Menger (1935), the lattice 
reformulation of both projective and affine geometry made 
logical use of Euclid’s definition in the deductive treatment of 
geometry. Euclid and his successors omitted to state what they 
meant by ‘part’; Menger supplied the deficiency by a precise 
logical definition. By corresponding explicit statement and 
inclusion of all hypotheses underlying his proofs, Menger 
showed that “great parts of projective and affine geometry 
may be developed upon the basis of ‘trivial’ axioms such as, 
for example, the postulates of [intersection and union as defined 
in Boolean algebra]. 

The outcome a mounted to the reduction of projective geo- 
metry to the alg'ebra of a suitably specialized lattice. The 
appropriate lattice 1 ? elements represent the several geometric 
objects or configurations of historical or mathematical interest, 
such as points, linehs, planes; the intersection of two configura- 
tions, in the Boolean sense, is the geometric intersection, or the 
configuration comntion to the two; the union of two configura- 
tions is the least 1 inclusive configuration containing both. 
Dcdekind’s axiom (ifor lattices, noted earlier), with a suitable 
finiteness condition t classifies the elements dimensionally by 
the corresponding Jordan-H older chains. In contrast to the 


previous axiomatizations by Hilbert and by Yeblen and Young, 
the lattice representation distinguishes no configuration as 
basic; all enter the theory symmetrically. To include affine 
geometry, Mengcr (1935) imposed on lattices a reasonable axiom 
of parallelism. He was thereby enabled to develop projective 
and affine geometry together. 

In G. BirkhofFs reduction (1934), projective geometries 
were correlated with Boolean algebras, fields and rings of point- 
sets, systems of normal subgroups, systems of ideals, modules 
of a modular space, and systems of subalgebras of abstract 
algebras. There were also further correlations with the reduction 
of group representations, semi-simple hypercomplcx algebras, 
and compact Lie groups. All of these were to be expected from 
the reduction of projective geometries to lattices of certain 
types, and from the earlier instances of lattices given mostly 
by G. Birkhoff himself. 

From a philosophical point of view, perhaps the most 
interesting consequence of the lattice (or structure) represen- 
tation of projective and affine geometries was a possible re- 
percussion on the perennial speculations of metaphysicians 
concerning the nature of space. The debate of the lS20’s on the 
real spacial existence of the ideal elements of projective geom- 
etry has been mentioned, and it was suggested that the 
disputed topic had no meaning in the sense intended by the 
debaters. The intimate connections between the Boolean 
algebra of logic, projective and affine geometries, and lattices 
at least hint that the debaters may have been discussing merely 
the ingrained habits of their own reasoning processes, inherited 
for thousands of years as a legacy from the two-valued logic 
in which, it appears, human beings reason with a minimum ol 
thought. ‘Space’ may be nothing more mysterious than a 
triviality of a rudimentary logic. 

Retrospect and prospect 

Having rapidly traveled one main highway of modern mathe- 
matics as far as 1945, we may now glance back over our route 
and note certain landmarks of more than local — arithmetical 
or algebraic — interest. This has indeed been our objective from 
the start: to observe those changes in spirit and point of view 
which distinguish the whole of mathematics since 1900 from that 
of the nineteenth century. 

Vast empires of mathematics, where hundreds of assiduous 


workers have toiled and where many still labor, have not been 
even noticed. A detailed survey of the territory traversed would 
fill many volumes but, so far as may be judged at present, a 
complete account would neither add to nor subtract from two 
conclusions, evidence for only one of which has so far been 
presented. Both may be falsified tomorrow by the advent of 
another Descartes or a modern Gauss, or by a successor of Galois 
or Abel. 

The mathematics of the twentieth century differs chiefly 
from that of the nineteenth in two significant respects. The first 
is the deliberate pursuit of abstractness, in which relations, not 
things related, are the important elements. The second is an 
intense preoccupation with the foundations on which the whole 
intricate superstructure of modern mathematics rests. It maybe 
hazarded as a very problematical guess that when the history 
of mathematics is written a century hence, if mathematics is to 
last that long, the early twentieth century will be remembered 
chiefly as the first great age of healthy skepticism in mathe- 
matics as in much else. 

The nineteenth century in mathematics seems in retrospect 
to be of a piece with the rest of that smugly optimistic age. God 
was in his Heaven and all was right with the world. European 
civilization shipped its blessings wholesale to the heathen of all 
continents. There seemed to be no limit to the quantity of 
flimsy goods that could be manufactured and dumped at a 
handsome profit on the unenlightened who w r ere unable to dis- 
tinguish tin plate from sterling silver, or brass nose rings from 
solid gold. Nor v r as there any restraint in the output and con- 
sumption of mathematics. Nearly everybody appears to have 
believed that nearly everything was sound beyond all doubt. 
With the turn of the century a period of criticism and revalu- 
ation set in, and all but unteachable reactionaries agreed that 
the change was a decade or more overdue. 

The origins of both the abstract method and the critical 
approach can be traced definitely to the 1880’s. Neither attracted 
much attention till Hilbert in 1899 published his work on the 
foundations of geometry, and about the same time pointed out 
the basic importance for all mathematics of proving the self- 
consistency of common arithmetic. But it seems just to attribute 
the initial impulse to G. Peano (1858—1932, Italian) in his postu- 
lates for arithmetic (1889). Resuming the Euclidean program, 
Peano undertook the deduction of common arithmetic from an 


explicitly stated set of postulates which were as free of con- 
cealed assumptions as he could make them. The postulational 
method is the source of both the modern critical movement and 

In the least flattering light, both criticism and abstraction 
reflect the leaden hue of decadence. Viewed thus, the mathe- 
matics of the twentieth century is a modernized version of the 
Alexandrian age of criticism and sterile commentary that were 
the lingering death of Greek mathematics. Even if this should 
prove to be a correct diagnosis of twentieth-century mathe- 
matics, it docs not necessarily follow that mathematics is about 
to expire. Although they were long in coming, Archimedes had 

Looking more sympathetically at the mathematics of the 
twentieth century, as nearly all professionals do, we see it full of 
life and more vigorous than ever. Criticism is necessary to see 
exactly what is sound so that the next step may be taken with 
reasonable safety. The abstract, postulational method is not 
mere cataloguing and pigeon-holing. It also is creation, but of a 
kind more basic than the disorderly luxuriance of the nineteenth 
century. Unless the enormous accumulations from that most 
prolific century in mathematical history arc sorted out and 
reduced to manageable proportions, mathematics will be 
smothered in its own riches. In the process of putting order into 
the huge mass by the abstract method, it is seen that much can 
be ignored. Should any of these neglected acquisitions ever be 
required, they arc now obtainable with much less labor than 
formerly. On the creative side, the postulational analysis of 
mathematical systems suggests innumerable new problems, 
some of which may be worth detailed investigation. 

Probably few engaged in this work of revaluation, simplifi- 
cation, and generalization imagine that an end has been reached 
in any direction. If Dedekind’s dual groups passed all but 
unnoticed for over thirty years, is it credible that all the promis- 
ing approaches have been explored, or that others will not be 
unexpectedly come upon ? A historical detail which was omitted 
in our rapid survey may be recalled here to suggest at least the 
possibility of progress in directions not yet followed. 

One of the clues that led Dcdekind to his creation of ideals 
was the theory of composition of binary quadratic forms, which 
is reflected in the multiplication of ideals in a quadratic field. 
Gauss systematized this theory of composition, and had cssen- 


L. J. Mordell (1888-, U.S.A., England) proved (1942) that the 
preceding equation has no solutions with a-, y, z, quartic poly- 
nomials in one parameter with rational coefficients unless 
it = a 3 or n — 2a z , where a is a rational number; and B. Segre 
(Italian, England) made (1943) extensive applications of alge- 
braic geometry to polynomial diophantine equations. 

There is the possibility, however, that the problem of arith- 
metical forms will lose its interest for mathematicians. In other 
words, it will cease to be considered important. But importance 
sometimes, humanly enough, is only a tribute to the self- 
esteem of an egocentric mathematician. The problems which he 
can solve are, by definition, important; those which baffle him 
arc unimportant. To say that a particular problem has lost its 
importance for modern mathematics may therefore be merely a 
rationalized confession of incapacity. Until a method has been 
devised for solving a problem, or for proving that it is unsolvable 
if such be the case, common professional pride would seem to 
demand that it be considered. 

If this point of view is justified, the conclusion is that neither 
algebra nor arithmetic has reached an end in the modern ab- 
stract method, although that method may be a significant pre- 
lude to what our successors will create. We shall see that physics 
suggests the like in analysis. In any event, the beautiful achieve- 
ments of the method would have delighted Euclid, who first of 
all mathematicians produced a rounded example of the postula- 
tional technique. He may not have realized what he was doing, 
as a modern mathematician sees his work, but he did it. Pythag- 
oras, on the other hand, would have stood bewildered before the 
modern concept of number. lie would want to know what had 
become of the natural numbers 1, 2, 3, ... . All this time wc 
have been taking them for granted, not questioning their 
specious simplicity. 

Wc must now return to these so-called natural numbers — 
Kroneckcr’s gift from God — and sec what happened to them 
while number was disporting itself in a heaven Pythagoras 
never dreamed of. This will supply us with the link connecting 
algebra and arithmetic with analysis. Having examined this, we 
shall be in a position to proceed in later chapters to geometry 
and applied mathematics. Finally we shall be led back to the 
foundations of the whole structure, to see in them what our 
successors may designate as the characteristic distinguishing 
the mathematics of the twentieth century from all that pre- 
ceded it — critical, constructive doubt. 


Cardinal and Ordinal to 1902 

Lecturing in 1934—5 to his Chinese students in Peiping on 
the theory of functions of real variables, a distinguished Ameri- 
can analyst 1 observed that “The student has thus far taken the 
system of real numbers for granted, and worked with them. He 
may continue to do so to the end of his life without detriment 
to his mathematical thought. . . . On the other hand, most 
mathematicians are curious, at one time or other in their lives, 
to see how the system of real numbers can be evolved from the 
natural numbers.” The natural numbers are the positive ra- 
tional integers 1, 2, 3, .... A little earlier, a distinguished 
German analyst, 2 writing for his beginners in analysis, made an 
unusual request: “Please forget all that you have learned in 
school; because you have not learned it.” He was referring to 
such simple matters as 1 + 1 =2. 

Subscribing to all of these sentiments, we shall indicate the 
main steps by which mathematicians in the second half of the 
nineteenth century reached the modern concept of real numbers. 
The real numbers are the soil in which the classical theory of 
functions grows and flourishes and, as remarked above, are 
usually taken for granted by students. Also by some others. 

To anticipate, theories were constructed deriving the real 
numbers from the natural numbers. It was then sought to de- 
rive the natural numbers from something yet more basic, the 
theory of classes as in mathematical logic. By another of those 
curious coincidences regarding dates, what seemed like finality 
in this direction was reached in the closing years of the nine- 
teenth century. Those years, in more senses than one, were the 
end of a great epoch. 




It is still no doubt true that students may take the real 
numbers for granted, as they did in 1902, without detriment to 
their mathematical thought. But it is no longer true that the 
more basic natural numbers can be taken for granted by any- 
body as they were by nearly everybody in 1902. Drowsy intui- 
tion has been shocked awake since the close of the nineteenth 
century; and the program of Eudoxus, resumed in the late nine- 
teenth century by the founders of the modern real number 
system, gave way in the twentieth century to another, more 
fundamental than any imagined by the Greek mathematicians. 
The center of interest shifted here as it is always shifting in 
mathematics. Indeed, it might have been mathematics and not 
an insignificant minor planet of a second-rate sun that Galileo 
had in mind when he muttered — according to a legend which 
should be true even if it may not be — “And still it moves,” as he 
rose from his knees and bowed to the Grand Inquisitor. Nobody 
yet has succeeded in stopping mathematical progress, in 1902 or 
in any other climactic year, as Joshua stopped the motion of a 
heavenly body at the battle of Gibeon. 

Equivalence and similarity 

The modern number concept is the link connecting the 
arithmetic and algebra of the past with the analysis, geometry, 
and mathematical logic of the present. Like the typical student 
of analysis today, we have been taking for granted the system 
of natural numbers from which, by successive generalizations, 
evolved the system of complex numbers, which in its turn sug- 
gested the hypercomplex numbers of modern algebra. Our im- 
mediate concern in this chapter is to indicate the main steps by 
which mathematicians in the latter half of the nineteenth cen- 
tury sought to ‘arithmetizc’ analysis. In the following chapter, 
we shall reach the same goal by a different route, that of the 
calculus from Newton and Leibniz to the year 1900; and we shall 
see again that the close of the nineteenth century marks a 
definite terminus in one direction of mathematical thought. 

This terminus, as will appear in the concluding chapter, v/as 
a turning point in the evolution of mathematics comparable in 
significance to that in the fourth century b.c., when Eudoxus 
parted company with the Pythagoreans. Once more the occasion 
for the new departure was the nature of irrational numbers. 
But in the modern pursuit of number, a far deeper spring of 
mathematical knowledge than any that refreshed the Greeks 


was tapped. The natural numbers had been taken for granted 
and it was found that they too presented subtle obstacles to 
clear understanding. The point of cardinal importance to be 
noted in the following summary account is the precise nature 
of the subtlest of these obstacles, which was perceived with 
dramatic suddenness only in 1902. 

The modern attack on number was directed against two 
closely related objectives: that of rigorizing the concepts of 
function, variable, limit, and continuity in analysis; that of 
penetrating the logical disguise of number. The first eventuated 
in the retreat from intuitive ideas of the calculus sublimated 
from unanalyzed conceptions of motion and continuous curves; 
the second culminated in the identification of cardinal numbers 
with classes. In both, the concepts of equivalence (or similarity) 
of classes, especially for infinite classes, played a dominant part. 
It is a matter of great historical interest (as indicated in an 
earlier chapter), that equivalence for classes was firmly grasped 
as early as 1638 by Galileo, 3 just a year after Descartes published 
his geometry. Galileo’s work was translated into English in 
1665, the year that young Newton, rusticating at Woolsthorpe, 
thought out his first calculus. 

To us it seems strange that so plain an indication as Galileo’s 
of a feasible attack on all matters pertaining to the infinite was 
not pursued sooner than it was. But there is the earlier parallel 
of the Greek indifference to Babylonian algebra to suggest that 
mathematics does not always follow the straightest road to its 

As it would be difficult to find a clearer, more graphic state- 
ment than Galileo’s of the critical points, we quote 4 what he puts 
into the mouths of his characters, the sagacious Salviatus 
( Salv .), and the questioning Simplicius (Simp.). The talk has 
been about the “Continuum of Indivisibles.” 

Salv. ... an Indivisible, added to another Indivisible, produceth not a 
thing divisible; for if that were so, it would follow, that even the Indivisibles 
were divisible. . . . 

Simp. Here already riseth a doubt, which I think un re solvable. . . • 
Now this assigning an Infinite bigger than an Infinite is, in my opinion, a 
conceit that can never by any means be apprehended. 

To make the infinite plain even to Simplicius, Salviatus 
patiently explains what a square integer is before proceeding as 



Salv. Farther questioning, if I ask how many are the Numbers Square, 
you can answer me truly, that they be as many, as arc their propper roots; 
since every Square hath its Root, and every Root its Square, nor hath any 
Square more than one sole Root, or any Root more than one sole Square. 

This is the kernel of the matter: the one-one correspondence 
between a part of an infinite class, here that of all the natural 
numbers, and one of its subclasses, here that of all the integer 
squares. Continuing the argument, Salviatus compels Simplicius 
to surrender. 

Simp. What is to be resolved on this occasion? 

Salv. 1 sec no other decision that it may admit, but to say, that all Numbers 
arc infinite; Squares are infinite; and that neither is the multitude of Squares 
less than all Numbers, nor this greater than that: and in conclusion, that the 
Attributes of Equality, Majority, and Minority have no place in Infinities, 
but only in terminate quantities. . . . 

In modem terminology, two classes which can be placed in 
one-one correspondence are said to be equivalent 1 or similar. 5 
In Galileo’s example, the class of all square integers is equivalent 
to the class of all positive integers. Again, a part (strictly, proper 
part) of a class C is any class which contains some but not all 
members of C, and nothing else. Galileo’s example shows that a 
class may be equivalent to a part of itself. A class is defined to 
be infinite if it is equivalent to a part of itself; and a class which 
is not infinite is defined to be finite. This means of distinguishing 
between finite and infinite classes was postulated 5 by B. Bolzano 
(1781-1848, of Prague), philosopher and theologian, and is 
basic in the modern theory of classes, both finite and infinite. 
Without it, Cantor’s theory of sets of points, fundamental in 
modern analysis, would not exist. 

It is interesting to note that Leibniz pointed out the simi- 
larity of the classes of all natural numbers and all even natural 
numbers, but drew the incorrect conclusion, rectified in Cantor’s 
theory, that “the number of all [natural] numbers implies a 
contradiction.” 7 

Arithmetized analysis 

This is not the place to expound the theories of real numbers 
constructed by Cantor, Dcdckind, and Wcicrstrass; and we shall 
merely recall the few fundamental concepts necessary to give 
point to the historical climax of 1902. Wc observe first that the 
concept of a class (set, aggregate, assemblage, ensemble, Mcnge) 
was taken as intuitive in the preceding section. That ‘class’ is 


by no means an intuitive notion was recognized by Cantor, who 
in 1895 defined it thus: “By a class (Menge) we understand anv 
summary (Zusammenfassung) into a single whole of definite 
well-distinguished objects of our intuition (Anschauung) or of 
our thought (Denkens).” Possibly the exquisitely modulated 
philosophical German 5 is untranslatable into any blunter lan- 
guage, or incomprehensible to any but the initiated. With hesita- 
tion, then, we offer the following substitute in crude American ? 
“A class is said to be determined by any test or condition which 
every entity (in the universe considered) must either satisfy or 
not satisfy.” It seems clear to the unphilosophic mind that 
either of these definitions somewhat rashly invites philosophers 
to philosophize; and indeed, the invitation was accepted with 
alacrity. Whether this was a happy issue for the analysis of the 
nineteenth century out of all its afflictions seems to be in some 
doubt among professional analysts. 10 

Another fundamental point in Cantor’s theory is the radical 
distinction between cardinal and ordinal numbers. For finite 
classes and numbers the distinction is almost trivial. Finite 
classes have the same cardinal number if and only if they are 
similar. Note that this does not define ‘cardinal numbed; it 
defines ‘same cardinal number,’ a significant distinction. It is 
quite possible to know that two criminals have the same name 
without knowing what the name is. The symbol 1, or 2, or 
3, . . . denoting a cardinal number (not yet defined!) of a class 
is a mere mark or tag which is characteristic of the class without 
reference to the order in which its members are arranged. When 
the members of a finite class are counted in a given order, the 
mark or label 1 being assigned to the first, the mark 2 to the 
next, and so on, an ordinal number is correlated with each ele- 
ment of the ordered class; and if n is assigned to the last, n is also 
the mark denoting the cardinal number of the class. But ior 
infinite classes, as shown by Cantor, the like is no longer true; 
the marks for (transfinite) cardinals and ordinals differ, and the 
distinction between cardinal and ordinal is not trivial. 

The cardinal number of anv given class, finite or infinite, was 
defined by F. L. G. Frege (1848-1925, German) to be itself a 
class, namely, the class of all those classes similar to the given 
class. Thus the familiar cardinals 1, 2, 3, . . . of our unleame 
youth have vanished in the aO-nesses of infinities of classes con- 
taining respectively ‘one’ thing, ‘two’ things, and so on ior as 
many things as there may be in our Anschauung or in our 



Dcnkcns. This outcome may seem rather disappointing at first. 
But on prolonged reflection we are forced to agree with E. 
Landau (1877-1938, German) that what we learned at school 
we did not learn. Any class similar to the class of all the natural 
numbers is said to be denumerable or countable. 

Resolving Simplicius’ doubt about the conceit of “assigning 
an Infinite bigger than an Infinite,” Cantor proceeded to describe 
any desired number of such bigger Infinites. First, there is said 
to be no difficulty in imagining an ordered infinite class; the 
natural numbers 1, 2, 3, . . . themselves suffice. Beyond all 
these, in ordinal numeration, lies co; beyond cs lies w + 1; then 
u + 2, and so on, until co2 is reached, when w2 -j- 1, ufl -f 2, 
. . . are attained; beyond all these lies to 2 , and beyond this, 
or + 1, and so on, it is said, indefinitely and forever. If the first 
step — after which all the rest seems to follow of itself — offers 
any difficulty, we have but to grasp the scheme 1, 3, 5, , 

2« + 1, . . . |2, in which, after all the odd natural numbers 
have been counted off, 2, which is not one of them, is imagined 
as the next in order. One purpose of Cantor in constructing these 
transfinitc ordinals, os, to + 1, . . . was to provide a means for 
the counting of well-ordered classes, a class being well-ordered 
if its members are ordered and each has a unique ‘successor.’ 

For cardinal numbers also Cantor described “an Infinite 
bigger than an Infinite” to confound the Simpliciuses of mathe- 
matics and enchant the Salviatuses. He proved (1874) that the 
class of all algebraic numbers is denumerable, and gave (1878) 
a rule for constructing an infinite non-denumcrabic class of real 
numbers. Were we to make a list of the spectacularly unexpected 
discoveries in mathematics, these two might head our list. 
Cantor’s proof is strictly one of existence. Providing no means for 
constructing any of the infinity of transcendental numbers whose 
existence is demonstrated, Cantor’s proof is in the medieval 
tradition of submathcmatical analysis. It would have convinced 
and delighted Aquinas. J. Liouville (1809-1S82, French), on the 
other hand, invented a method (1844) for constructing any one 
of an extensive class of transcendental numbers. His numbers 
were the first to be proved transcendental; Hcrmitc’s proof of the 
transcendence of e( — 2.718 . . .) followed in 1S73; F. Lindc- 
mann’s (1852-1939, German) forrr in 1882. Hinting at the con- 
troversies to come in the twentieth century, Kronecker 
demanded of Lindcrnann, “Of what value is your beautiful 
proof, since irrational numbers do not exist?'' We shall return to 


KroneckeP s program of arithmetization presently. It was quite 
different in both aim and scope from that of Cantor, Dedekind 
and Weierstrass in their project of arithmetizing analysis. In 
passing, we note that A. Gelfond in 1934 proved the transcen- 
dence of a b , where a is any algebraic number ^0, 1, and b is any 
irrational algebraic number. 

In the program of arithmetizing analysis, the rational 
numbers presented no difficulty. By the device of number- 
couples subjected to appropriate postulates, the properties of 
positive rationals were referred to those of positive integers, and 
negative rationals were driven back with equal ease to positive 
rationals. Thus all the rationals were derived by a simple routine 
from the natural numbers. Proceeding to infinitely the greater 
part of the continuum of real numbers, Cantor defined irrationals 
by infinite sequences of rationals; for example, may be 
defined by the sequence 1, 14/10, 141/100, 1,414/1,000, 
14,142/10.000, .... Generally, if a i} a~, a z , . . . is any infinite 
sequence of rational numbers which is such that, for each 
rational e > 0, however small, there is an index m such that 
\a n — a\ < e for every n , v Se m, the sequence is said to be 
regular. It is postulated that every regular sequence defines a 
number; the class of all so-defined numbers is the real number 
system. With suitable definitions of equality', greater, less, sum, 
difference, product, and quotient, it was shown that these num- 
bers satisfy the requirements of experience. In particular, a 
meaning was given to the formalism of such useful statements as 
V2 X A/3 = V2 X 3, V2 X VI = V3 X VI. Cantor had arith- 
metized the continuum of real numbers. 

Geometry also shared in the benefits which arithmetization 
had conferred on analysis. A one-one correspondence between 
all the points on any segment of a straight line and the con- 
tinuum of real numbers was established. This done, C. Jordan 

(1838-1922, French) banished intuition from the conception of 
curved lines by giving a strictly arithmetical definition 11 oi a 
curve as a plane set of points which can be put in one-one 
correspondence with the points of a closed segment [a, b). This 
seems like restating a platitude in pedantic obscurity when we 
cite its simplest example, the parametric equations x = r cos 1, 
y = r sin t of the circle x" -r y~ = t~. It seems rather less plati- 
tudinous when we recall that Peano (1890) constructed a re^ 
continuous plane curve, as the locus of a point {x, y) >’hose 
~ _ if A = g( t ) y-ith /, g uniform- 

coordinates are given by x 



continuous functions of the real variable t in the range 0 g / g 1, 
completely filling the square 0 g x ^ 1, 0 y g 1. In fact, he 
described two such curves passing through every point of the 
unit square. Many more examples of such ‘space-filling curves’ 
have been constructed since Peano exhibited the first; and what 
in 1890 appeared as a collapse of the geometrical heavens has 
reappeared as a commonplace phenomenon in Ph.D. disser- 
tations. “And still it moves.” 

Equally unexpected miracles began illuminating the con- 
tinuum itself. By a fairly immediate gencraliaation, classes (sets) 
of points in a continuum (‘space’) of any finite or of a denumer- 
ably infinite number of dimensions were invented. Cantor proved 
that in each instance all the points in the whole space can be put 
in one-one correspondence with all the points on any straight-line 
segment. In a plane, for example, there are precisely as many 
points on a segment an inch long as there are in the entire plane. 
This, of course, is contrary to common sense; but common sense 
exists chiefly in order that reason may have its Simpliciuses to 
contradict and enlighten. However, some Simplicius occasionally 
interjects a shrewd objection, upsetting the even progress of the 
discussion; and if he but seldom gets the better of an argument, 
he can at least cause it to stumble badly. Kronecker elected 
himself the Simplicius of Cantor, Dcdekind, and Weierstrass. 
His objections will be noted presently. 

A profound question which exercised Cantor’s utmost 
powers was this: Can the continuum of real numbers be well 
ordered? In 1883 he thought he had answered this affirmatively. 
Objections to his attempted proof were largely responsible for 
the descent of mathematics after 1900 to deeper levels in its 
efforts to escape deceptive intuition. Another problem which 
baffled Cantor was to prove or disprove that there exists a class 
whose cardinal number exceeds that of the class of natural 
numbers and is exceeded by that of the class of real numbers. 
’1 his problem seemed to be still open ,: in 1945. 

Whatever may be the ultimate fate of Cantor’s theories of the 
infinite, continuity, and the number system, it appears likely 
that he will be remembered with Eudoxus as one of those who 
breached what is, after all, the central fortress of mathematical 
analysis. So also will Dcdekind and Weierstrass. Like Cantor, 
these two also derived the number system of analysis from the 
natural numbers. Dcdekind by his device of cuts, Weierstrass by 
classes of rationals. Another analyst who reached the same goal 


was C. Mera y (1835-1911, French); but possibly the difficulty of 
his exposition deprived him of his just share of fame. One or 
other of these theories is so familiar today to every student of an 
advanced course in the calculus that there is no need to describe 
them here. They are open to precisely the same objections that 
mathematical logicians have raised to Cantor’s Mengenlehre. 
But in stating this plain matter of fact, we do not imply that 
these theories have been rejected as totally wrong or barren. 
They still afford the most promising approach to an under- 
standing of numbers and their part in analysis. If a theory is 
imperfect, that is perhaps only because it is not yet either dead or 

In the following chapter we shall see that the analysis of 
trigonometric (Fourier) series was partly responsible for the 
attempt to put a firm logical foundation under the continuum of 
real numbers. Many contributed significantly to the attempt; 
but the four whose work has been noted were the earliest to see 
clearly what needed doing and the first to attempt it. 

Of the many who prepared for the final success, one may be 
mentioned here, P. du Bois-Reymond (1831-1889, German), 
partly for his own subtle researches in analysis, partly because 
it was owing to his insistence that Weierstrass permitted a most 
disconcerting invention of his own to become public. Intuitively, 
a continuous arc of a curve has a tangent at every point on the 
arc; Weierstrass constructed the equation of a continuous curve 
having no tangent at any of its points. He is said to have com- 
municated this to his circle in 1861, but for some reason withheld 
it until du Bois-Reymond in 1874 asked him whether such a 
curve was possible. This example alone showed the necessity for 
some rigorous theory of the real number system to replace the 
pernicious intuitions which had seeped into analysis from 
geometry and kinematics. 

Existence and constructibility 

Kronecker has already been mentioned for his technical con- 
tributions to algebra and the higher arithmetic. These probably 
represent his choicest creations to those who have taken the pains 
to appreciate them; but Kronecker is more widely known to the 
mathematical public for his philosophy of mathematics. At one 
time he was looked upon by some of the analysts, including 
Weierstrass, as a sort of personal devil. It was feared that 
Kronecker’s philosophy was wholly destructive; and it cannot 


be gainsaid that he hated the highly speculative analysis of his 
famous contemporaries. If Kronecker spelled Satan to Cantor, 
Cantor signified the personification of all mathematical evil to 

Dcdckind’s definition of irrationals as cuts in infinite classes 
of rationals, Cantor’s sequences of rationals defining irrationals, 
and Weierstrass’ irrationals as classes of rationals, all ultimately 
referred the continuum of real numbers to the natural numbers. 
The ‘magnitudes’ of Eudoxus were replaced by hypothetical 
constructions performed upon the numbers 1, 2, 3, ... . Thus 
the arithmetization of analysis was a return to the program of 
Pythagoras. Mathematical mechanics having been reduced to a 
department of analysis, it too was potentially arithmetized, at 
least by implication, and likewise for geometry. All at last was 
reduced to number as the Pythagoreans imagined number, but 
at a cost which they would never have attempted to meet, 
infinities upon infinities. 

As thoroughgoing a Pythagorean as Pythagoras himself, 
Kronecker insisted that the infinities be banished, and that all 
mathematics be built up by finite constructions from the 
natural numbers. Unless a mathematical object were construc- 
tiblc in a finite number of non-tentative steps, it did not exist 
for Kronecker, no matter how many and how rigidly logical the 
transcendental proofs of its existence. 15 Such a philosophy of 
mathematics did not make nonsense of analysis as Kroncckcr’s 
rivals had re-created it; analysis was simply abolished. 

As if to lend some plausibility to the destructive part of 
Kroncckcr’s program, flagrant contradictions began appearing 
in the late 1890’s in reasoning of apparently the same general 
character as that used by the arithmetizers of analysis. Twenty- 
four centuries after Zeno’s runner had lost his race, his heirs 
appeared on another course, fresher and fleeter of foot than that 
ancient had ever been. The new antinomies of the infinite sprang 
from the protean ‘all’ by which the irrationals were generated 
in the arithmetization of analysis: ‘all’ natural numbers; the 
class of ‘all’ rational numbers whose squares arc less than 2, and 
the class of ‘all’ those whose squares arc greater than 2 as the 
‘cut’ defining VI, and an infinity more. 

The first of the new and more vigorous paradoxes was 
fathered in 1897 by the Italian mathematician C. Burali-Forti: 
the well-ordered scries of all ordinal numbers defines a nau 
ordinal number which is not one of the all. 


A less technical parados on ‘all’ was B. A. W. Russell’s 
(1872-, English) of 1902: Is the class of all those classes which 
are not members of themselves a member of itself? Either Yes 
or No leads to a contradiction. The same irrepressible successor 
of Zeno recorded a yet simpler ‘all’ paradox: A barber in a 
certain village shaves all those, and only those, who do not shave 
themselves; does the barber shave himself? There are many- 
more. All — if we may use the word without danger of engender- 
ing another exasperating paradox — or, if not all, then many 
conceal a dubious ‘all.’ One root of the technical mathematical 
difficulties has been traced to the concept of ‘class’ itself, 
Cantor’s ‘Menge’ as defined by him. The attempt to be logically 
precise ended in hopeless confusion. 

The problems of the nineteenth century met those of the twen- 
tieth in the extraordinarily subtle mind of F. L. G. Frege (1848— 
1925, German), part of whose lifework 14 was an endeavor to put 
a self-consistent foundation under the number concept. In 1884, 
Frege was led to his famous definition of the cardinal number of 
a given class as the class of all those classes similar to the given 
class. From this definition, Frege derived the usual properties of 
numbers familiar in common arithmetic. Unfortunately, to 
develop the subtlety of his reasoning with precision, he had 
found it necessary to clothe his proofs in a complicated dia- 
grammatic symbolism which repelled all but the hardiest and 
most obstinate readers. As a result, the epoch-making definition 
embedded in his work passed unnoticed by the mathematical 
public till Russell, by different reasoning, independently reached 
(1902 ?) the same definition and expounded it in English. 

Frege had used the theory of classes. The second volume of 
his masterpiece 14 appeared in 1903. It closes with the following 
confession. “A scientist can hardly encounter anything more 
undesirable than to have the foundation collapse just as the work 
is completed. A letter from Mr. Bertrand Russell put me in this 
position as the work was all but through the press.” Russell’s 
letter contained his paradox on the class of all classes that are 
not members of themselves. 

Frege’s pessimism is understandable enough. But it was to 
prove unjustifiable in the long view of mathematical progress. 
The attempt to found the number system on a theory of classes 
seemed to have failed, and no doubt it had, at least temporarily. 
By the collapse of the class theory of the number system, analysis 
was left without a foundation, and hung suspended in mid-air 



like Mahomet’s coffin, sustained only by a miracle of faith. But 
the very failure revealed the nature of the fundamental weak- 
ness. A younger, more vigorous generation attacked the problem 
of bringing analysis down to reason again. Profiting by the 
experience of the nineteenth-century arithmetizers, the mathe- 
matical logicians of the twentieth century set themselves the 
task of putting a self-consistent foundation under all mathe- 
matics, not merely under analysis. Their efforts quickly carried 
the program of Leibniz for a strict symbolic reasoning far beyond 
anything he ever conceived, and in so doing created much new 

In the meantime, analysts, geometers, arithmeticians, and 
algebraists continued their technical labors as if there were no 
‘crisis’ in the foundations, creating interesting and useful things 
as their forerunners had done for centuries. Their confidence in 
the security of the essentials of their creations is justified by 
experience. Changing philosophies of mathematics may trans- 
form proofs and even theorems out of easy recognition as mathe- 
matics develops, and much is thrown away. But if history is a 
reliable prophet, 15 there will remain of the analysis of the nine- 
teenth century as much, relatively, as remains of Euclid’s 
proposition I, 47. 


From Intuition to Absolute 


In following the development of the number concept to its 
final phase in modern arithmetic and abstract algebra, we have 
caught occasional glimpses of the spirit of mathematics as it has 
become since the dose of the eighteenth century. Similar pro- 
found changes appear when we observe analysis. We now return 
to the eighteenth century and note the first attempts to construct 
a logically sound differential and integral calculus. 

The contrast between what passed for valid reasoning then 
and what is now demanded is violent. Passing back to the eight- 
eenth century, we find ourselves in a dead world, almost in 
another universe. Some of Newton’s successors who strove to 
make sense out of the calculus are among the greatest mathe- 
maticians of all time. Yet, as we follow their reasoning, we can 
only wonder whether our own will seem as puerile to our succes- 
sors a century and a half hence. It is not a question here of the 
enduring things these famous men did with their analysis in 
applied mathematics, or even of the basic algorithms which they 
invented and which have also lasted. We are concerned solely 
with their avowed attempts to put consistent meaning into the 
analysis itself. 

Two decisive turning points 

We saw that Newton himself was dissatisfied with his account 
of the fundamental concepts of the calculus. The like hoi s or 



Leibniz, %vho half promised Huygens that some day he would re- 
turn to the beginning and set everything right. But he never did. 
After the passing of Newton and Leibniz, critics of both ap- 
proaches made themselves heard, and conscientious analysts, 
responding to legitimate objections, attempted to put a firm 
foundation under the calculus. Their efforts gradually disclosed 
the depth of the difficulties, and in the nineteenth century were 
partly responsible for the creation of vast new departments of 
mathematics, such as the theories of Dedekind and Cantor. We 
shall indicate the main stages in this extremely complex evolu- 
tion by which the calculus of 1700 developed into that of 1900. 
As wide a gulf separated the great analysts of the late eight- 
eenth century from those of the early nineteenth as severed the 
Pythagoreans from Eudoxus. After 1929 — the historic year in 
which the great depression began in the United States — another 
deep fissure opened up, cutting off retreat to the nineteenth 
century, apparently forever, when K. Godel re-examined the 
possibility of a consistency-proof for rational arithmetic. 

In following any account of the evolution of rigor in the cal- 
culus, it must be remembered that opinions on many unsettled 
points differ, sometimes widely. Further, it has been difficult for 
some to avoid reading their own more exact knowledge into 
work of their predecessors which, if taken at what seems its 
intended value, gives no hint that its authors were ever conscious 
of what later stood out as fatal defects. For example, the gener- 
ous J. lc R. d’Alembert (1717-1783, French) in 1770 ascribed 
to Newton a fully developed theory of limits that but few 
analysts today can detect in what Newton published. And last, 
before proceeding to details, we emphasize once more that in 
exhibiting the shortcomings in the work of the older analysts, 
there is no implication that perfection has been attained in our 
own. The mistakes and unresolved difficulties of the past in 
mathematics have always been the opportunities of its future; 
and should analysis ever appear to be without flaw or blemish, 
its perfection might only be that of death. 

Five stages 

The general trend from 1700 to 1900 was toward a stricter 
arithmetization of three basic concepts of the calculus: number, 
function, limit. More subtle questions concerning the meaning 
of ‘variable’ scarcely entered before the twentieth century'. In 
the period under discussion there were five well-marked stages, 


which may be easily retained by the names and dates of certain 
leaders in each. With the first are associated Thomas Simpson 
(1710-1761, English) in England, and G. F. A. l’Hospital 
(1661—1704, French) on the Continent. Euler (1707-1783, Swiss) 
represents the second stage; Lagrange (1736-1813) the third- 
Gauss (1777— 1855) and Cauchy (1789—1857) the fourth; and 
Weierstrass (1815—1897, German) the fifth. Euler is the culmina- 
tion of the almost wholly uncritical schools of Newton and 
Leibniz; Lagrange marks the earliest recognition by a mathe- 
matician of the first rank that the calculus was in a thoroughly 
unsatisfactory condition; Gauss is the modem originator of 
rigorous mathematics; Cauchy is the first modern rigorist to 
gather any considerable following; and Weierstrass, dying in 
1897, epitomizes the progress made in exactly one hundred years 
from the first publication (1797) of Lagrange attempting to 
rigorize the calculus. 

The golden age of * nothing ’ 

‘Formalism 5 in analysis means manipulation of formulas in- 
volving infinite processes without sufficient attention to con- 
vergence and mathematical existence. Thus the formal binomial 
theorem applied to (1 — 2 ) _1 gives 

-1 = 1+ 2-1-4 + 8 + 16+..., 

a meaningless result which did not astonish Euler, the great- 
est but not the last of the formalists. ‘Intuition 5 in analy- 
sis, as we shall use it here, means an unreasoning faith in the 
universal validity of what the senses report to the intellect 
concerning motion and geometrical diagrams. Newton was the 
greatest of the intuitionists in analysis, with the far more 
philosophical Leibniz a distant and therefore honorable second. 
(Both formalism and intuitionism in the mathematics of the 
twentieth century have different meanings, to be noted in the 
concluding chapter.) The direction of evolution in the calculus 
has been constantly away from formalism and intuition, al- 
though neither is yet extinct. 

The first and crudest stage is represented in England by the 
two editions (1737-1776) of Simpson’s classic Treatise on 
fluxions , in which 1 intuition flourishes freely and rankly. At- 
tempting to clarify Newton’s intuitive approach to fluxions 
through the generation of “magnitudes” by “continued mo- 
tion,” Simpson succeeded only in adding a deeper obscurity o 


behind a goodly part of these savage assaults on the work of 
eminent mathematicians. The shrewdest attack of all was 
delivered by a man who was not a mathematician and who made 
no claim to be, G. Berkeley (1685-1753, Irish): half-heir of 
Jonathan Swift’s Vanessa; at one time self-appointed apostle of 
culture to Bermuda but shipped by mistake to Newport, Rhode 
Island, where for three stagnant years (1728-31) he rusticated; 
later Bishop of Cloyne in his native Ireland; famed for his sub- 
jective idealism that out-idealized Plato, and immortal for his 
advocacy of tar water as a remedy for spiritual disorders and 
smallpox. A mind as keen as Berkeley’s was needed to expose 
once for all the subtle fallacies in Newtonian fluxions, and the 
sagacious bishop spared no logic in his withering attack. A 
philosophical amateur did what professional mathematicians 
had shown themselves either too partisan or too tender-minded 
to do. Although few professionals would admit that either 
corpse was dead, Berkeley slew both fluxions and ‘prime and 
ultimate ratios.’ 

Berkeley’s assault in his Analyst (1734) was not just another 
of the vulgar wrangles, like the controversy over priority in the 
calculus, that disfigure the career of the Queen of the Sciences. 
It was one of the ablest critiques the leading mathematicians of 
any period have ignored, possibly because it came from one who 
was not a member of their somewhat exclusive guild. For once a 
philosopher turned the tables on mathematicians by convicting 
the fluxionists of changing their hypothesis in the middle of an 
argument. Until Berkeley’s time, it had been supposed that this 
effective tactic in logomachy was an exclusive prerogative of 
dialecticians. Berkeley contended that substituting x + o for 
x in x n and letting o vanish in the final step, to get the fluxion of 
x n , is a shift in the hypothesis: “ . . . for when it is said let the 
increments be nothing, 4 or let there be no increments, the former 
supposition that the increments were something, or that there 
were increments, is destroyed, and yet a consequence of that 
supposition, i.e., an expression got by virtue thereof, is retained. ’ 
There were replies to this; but that which is unanswerable can- 
not be answered, and the controversy blew over, leaving scarcely 
a ripple on the muddy waters of mathematical analysis as they 
were in 1734. 

Berkeley’s criticisms were well grounded, but neither he nor 
they were taken seriously by the leading analysts of his time, 
and mathematics sought salvation in its own way. It is amusing 


to recall in passing that another question of salvation inspired 
Berkeley to his attack on fluxions. The full title of his work is 
The analyst: or, a discourse addressed to an infidel mathematician. 
Wherein it is examined whether the object, principles, and infer- 
ences of the modern analysis are more distinctly conceived, or more 
evidently deduced, than religious mysteries a7id points of faith. 
Only an Irish bishop who was also an idealistic philosopher could 
have conceived such a heroic project. It seems that Newton’s 
friend Halley, posing as a great mathematician, had proved 
conclusively to some deluded wretch the inconceivability of the 
dogmas of Christian theology. The converted one, a friend of 
Berkeley’s, refused the latter’s spiritual offices on his deathbed. 
This was in the very year that Berkeley became a bishop. 
Profoundly shocked by the soul-destroying savagery of “the 
modern analysis,” and mindful of his education in semicivilizcd 
Rhode Island, the good bishop went after the scalp of fluxions. 
He secured it; and the wretch who had been converted to 
infidelity by a nonsensical argument was avenged, although it 
may have been too late to save his soul. 

The triumph of formalism 

Euler’s almost total capitulation to the seductions of formal- 
ism is one of the unexplained mysteries in mathematics. Like 
Newton, Euler was aware that scries must ‘in general’ 5 converge 
if they arc to be practically useful, as in astronomy; but unlike 
Newton, he was unable to restrain himself this side of absurdity. 
Euler appears to have believed that formulas can do no evil; 
and so long as they continued to furnish their parent with ever 
new and more prolific variations of themselves, he encouraged 
them to increase and multiply, trusting no doubt that someday 
all their offspring would somehow be legitimized. Many of them 
have, and flourish today as lusty theories whose first bold steps 
were taken in several editions of three masterpieces of this most 
prolific mathematician in history: Jntroduclio in analysin infini- 
torum (174S); calculi diffcrcntialis (1755); Institu- 
tions s calculi integralis (176S-1794). 

The aim of the Introductio is to obtain by elementary means 
the utmost of what is so obtainable, but which is usually derived 
by the differential and integral calculus. The work is in two parts, 
an analytic and a geometric. Among the host of results are the 
expansions of the circular (singly periodic) functions, trans- 
formations of infinite products into infinite scries, and develop- 


ments into series of partial fractions. The last suggested one 
approach in the nineteenth century to elliptic (doubly periodic) 
functions. One chapter derives the basic formulas in the analytic- 
algebraic theory of partition of numbers. There are two heroes 
in this great drama of formalism, the expansion of f from the 
limit (Euler’s style) of (1 + x/n) n as n tends through positive 
values to infinity, and the cardinal formula of analytic trigo- 
nometry, e' x = cos x i sin x, i = ( — 1)*. Creative formalism 
such as this is responsible for the impatient criticism of ex- 
treme mathematical rigor as rigor mortis. 

The geometric part handles analytic geometry, both plane 
and solid, with equal freedom and complete mastery. The 
material includes special curves and surfaces, tangents and tan- 
gent planes, normals, areas, and volumes. 

Turning partly away from intuitionism, Euler abandoned 
geometry in the Institutiones. This work is remarkable for its 
exhibition of analogies between the infinitesimal calculus and 
the calculus of finite differences, and the use of the latter to 
approximate to results in the former. There is no hint of con- 
vergence, but slowly convergent series are converted in masterly 
fashion into others more rapidly convergent. Here also the usual 
formal parts of the differential and integral calculus are devel- 
oped in minute detail. One prophetic triumph of manipulative 
skill may be specially cited: Euler obtains the addition theorem 
for elliptic integrals as an exercise in differential equations. 

A function to Euler became a congeries of formal representa- 
tions transformable into one another by ingenious devices rang- 
ing from elementary algebra to the calculus. Glorying in the 
pragmatic power of his methods, Euler needed to see nothing 
absurd in his conception of the differential calculus as a process 
of determining the ratio of vanished increments. His differentials 
are first and last absolute zeros whose ratios by some incompre- 
hensible spiritualism materialize in finite, determinate numbers. 
As the usually courteous Lagrange observed, Euler’s calculus 
does not make sense. 

If the end ever justifies the means in analysis, Euler was 
justified. He sought beautiful formulas, and he found them in 
overwhelming abundance. But obviously the calculus could not 
continue indefinitely on the primrose path so happily followed 
by this boldest and most successful formalist in history. Even 
Euler caught an occasional whiff of the everlasting bonfire in 
the absurdities that floated up now and then from the pit just 


ahead of him. Others scented damnation more keenly than he, 
among them his friend d’Alembert. 

Best known for his principle in mechanics (1743), d’Alem- 
bert should be remembered also for having been the first (1754) 
to state 6 that “the theory of limits is the true metaphysics of 
the differential calculus.” That nobody in the eighteenth century 
carried out the implied program, or was capable of doing so, is 
beside the point; d’Alembert saw clearly that what the calculus 
needed was not more formulas but a foundation. He regarded 
Newton’s calculus of prime and ultimate ratios as a method of 
limits. Newton might have agreed with this had it been pointed 
out to him. 

Lagrange's remedy 

A new direction was taken by Lagrange in his ambitious 
Theoric dcs fonctions analytiques (1797, 1813), and his Calcul dcs 
f evictions (1799, 1806). These -were conscious attempts to escape 
from Euler’s conception of a function as a mere formula or al- 
gorithm, although Lagrange himself substituted another kind 
of formula, the power series, for the representation of all func- 
tions. His escape took him from one kind of formalism to 
another. Dissatisfied 7 with the efforts of all his predecessors 
and contemporaries, he rejected both infinitesimals and limits as 
being unsound, too difficult for neophytes, and, in the least 
complimentary sense, metaphysical. 

Lagrange was the leading mathematician of the eighteenth 
century and one of the greatest in history. Pie also was the first 
to restate Taylor's theorem with a remainder term. Keeping all 
this in mind we shall, if we have the least grain of caution in us, 
be extremely conservative in our estimates of current rigor when 
we remember what finally satisfied Lagrange. 

He based his calculus on the expansion of a function in a 
Taylor scries, assuming that, "by the theory of series,” 

}{x + h) — /(*) -p ah -f- bh 2 -p ch s + • • • • 

From this he convinced himself that if a s= "the derived 
function of /(.v),” then 2b — f'{x), where f"(x) — (f'(x))\ and 
so on, all, as he imagined, without benefit of limits. He points 
out 3 that anyone familiar with the usual form of the calculus will 
sec that/'(.v) is really df(x)/dx. But by what he has just deduced 
from "the theory of scries,” it is clear that df(x)/dx docs not in 
any way depend upon limits, prime and ultimate ratios, or in- 


finitesimals as formerly ;/'(#) is merely the coefficient of h in the 
expansion of f(x + h ) in ascending powers of h. Need more be 
said ? 9 

Gains to 1800 

The net gains in the eighteenth century appear to have been 
four. Berkeley disposed of fluxions and of prime and ultimate 
ratios. Euler produced a vast wealth of results by purely formal 
uses of the calculus; and so sure was his instinct for what was to 
remain valid that his work was the point of departure from 
which many of his more productive successors made some of 
their most significant advances. To cite only two instances, 
Gauss, Abel, Jacobi, and Hermite were indebted directly to 
Euler in their more rigorous work on the theta and elliptic 
functions; the Eulerian integrals suggested to Legendre, Gauss, 
and Weierstrass extensive developments in the theory of the 
gamma function. 

The third outstanding gain was d’Alembert’s demand that' 
the calculus be founded on the method of limits. The execution 
of this program had to wait for Cauchy (1821). The fourth gain 
was the hint implicit in Lagrange’s abortive attempt to generate 
the calculus from power series. Weierstrass, in his theory of 
analytic functions, carried out what might have been an eight- 
eenth-century program had Lagrange seen just a little more 
clearly what he was actually doing. 

Ridiculous interlude 

Formalism of a narrower sort than Euler’s reached its absurd 
climax in the period between Lagrange and Cauchy. Com- 
binatorial analysis, in the trivial sense of manipulating binomial 
and multinomial coefficients, and formally expanding powers of 
infinite series by applications ad libitum ad nauseamque of the 
multinomial theorem, represented the best that academic 
mathematics could do in the Germany of the late eighteenth 
century. The combinatorial school headed by C. F. Hindenburg 
(1741-1808, German) was the unlovely offspring of two human 
failings, neither of which is popularly supposed to have any 
relevance for the sublimities of pure mathematics: blind hero 
worship and national jealousy. 

The German Leibniz in his combinatorial analysis of the dis- 
crete had created a rival for the English Newton’s infinitesimal 
analysis of the continuous. Therefore, abandoning the calculus 


and its astronomical applications to the British, the Swiss, and 
the French, who had the lead already, the German mathemati- 
cians would loyally follow their national hero. Completely miss- 
ing the deeper significance of Leibniz’ program as a step toward 
the ‘universal characteristic,’ his patriotic disciples elaborated 
its superficialities in a wealth of useless formulas. The title of the 
masterpiece in this ambitious futility brazenly proclaimed the 
multinomial theorem to be the most important truth in the whole 
of analysis. 13 

Still more grandiose pretensions to omnipotence were urged 
by an egocentric Pole, H. Wronski (177S-1S53), ardent envicr of 
Lagrange, also a disciple of the combinatorial school, although 
his transcendent conceit 11 denied any progenitor but Wronski 
for himself and his “Supreme Law” which, he insisted, con- 
tained all analysis, past, present, and future. Both Wronski’s 
claims and those of the combinatorialists have been disallowed 
by the supreme court of mathematical progress, from which there 
is no appeal. His criticism of Lagrange’s attempt at rigor was 
justified; but his own substitute was no better. 

The labors of this almost forgotten combinatorial school, 
however, were not without lasting benefit for the calculus. They 
filled young Gauss with such an intense disgust for formalism 
and all its works that he resolved to go his own lonely way and 
put some meaning into analysis, even if it cost him the patronage 
of every academician in Germany. He even favored the mighty 
Hindcnburg with an extremely sarcastic letter. 

In 1S12 Gauss published his classic memoir 1 - on the hyper- 
geometric scries, in which, for the first time in the history of 
mathematics, the convergence of an infinite series was ade- 
quately 1 ' 1 investigated. Others before Gauss had gone as far as 
stating tests for convergence, notably Leibniz, for alternating 
series, and E. Waring (1734-179S, English), who had given what 
is usually called Cauchy’s ratio test as early as 1776; but Gauss 
was the first to carry through a rigorous treatment. 

Intuition transformed 

From the calculus of Newton and Leibniz to that of Lagrange 
there is no indication that analysts were aware of the necessity 
for an understanding of the real number system. Nor is there in 
the next stage, that of Cauchy. Even as late as 1945 ‘quantities’ 
occurred frequently in the writings of professional analysts with 
no explanation of what a ‘quantity’ may signify. 


Imagining, perhaps, that he was banishing deceptive intui- 
tion forever from analysis, Cauchy succeeded in driving it down 
to a far deeper level where it might continue its subtle mischief 
unobserved. The crude visual and geometrical intuition of the 
early analysts was transformed into an uncritical faith in the 
logical possibility of the continuum of real numbers. Cauchy, 
Abel, and possibly Gauss 16 — for he seems to have left no record 
of his beliefs on this matter — adhered to this faith. 

The definitions of limit and continuity current today in 
thoughtfully written texts on the elementary calculus are sub- 
stantially those expounded and applied by Cauchy in his lectures 
and in his Cours d’ analyse (1821), his Resume des legons donnees 
a Vecole -polytechnique (1823), and his Applications du calcul 
infinitesimal a la geometrie (1826). The differential quotient, 
or derivative, is defined as the limit of a difference quotient, the 
definite integral as the limit of a sum, and differentials as arbi- 
trary real numbers. The continuity of a function and the con- 
vergence of an infinite series are referred to the concept of a 
limit. Thus Cauchy in effect created the elements of the classical 
theory of functions of a real variable. It was Cauchy’s rigor that 
inspired Abel on his visit to Paris in 1826 to make the banish- 
ment of formalism from analysis a major effort of his projected 

But, indicative of the subtleties inherent in consistent think- 
ing about the infinite and the continuum, even so cautious a 
mind as Cauchy’s went astray when it surrendered itself to 
intuition. He believed for a time that the sum of any convergent 
series of continuous functions is continuous, and that the integral 
of the sum is always obtainable by termwise integration. Later 
(1853, 1857) he recognized uniform convergence, discovered 
independently by the mathematical physicist G. G. Stokes 
(1819-1903, Irish) in 1847 and P. L. v. Seidel (1821-1896, 
German) in 1848. Cauchy also fell foul of the traps guarding 
interchange of limits in double-limit processes, as also did 
Gauss, 10 another plain hint that the real number system is less 
innocuous than it appears to naive intuition. 

A suggestion from physics 

It is rather surprising to find a main source of modern ngor 
in the work of a mathematical physicist who had almost a con- 
tempt 17 for mathematics except as a drudge of the sciences. 
J. B. J. Fourier (1758-1830, French) published his masterpiece, 


La thcoric analytiquc de la chaleur, in 1822, the year after Cauchy 
had rigorizcd the calculus. But if it had appeared twenty years 
after Cauchy’s lectures, it probably would not have differed 
materially from what it was. Fourier had obstinately refused for 
fifteen years to heed the objections of Lagrange and others that 
vital parts of his analysis were unsound. In his famous classic on 
the conduction of heat, 17 Fourier proved himself the Euler of 
mathematical physics. Leaving convergence to take care of 
itself, he trusted his physical intuition to lead him to correct 
results, as it usually did. 

The sixth section of Fourier’s Thcoric 17 is the one which con- 
cerns us here. It is devoted to the solution “of a more general 
problem, which consists in developing any function whatever in 
an infinite series of sines or cosines of multiple arcs. . . . We 
proceed to explain the solution.” 13 Having done so for a special 
case, Fourier continues, 15 “We can extend the same results to 
any functions, even to those which are discontinuous and entirely 
arbitrary'. To establish clearly the truth of this proposition, 
we must examine the foregoing equation,” which he does, in the 
manner of Eulcrian formalism. 50 The outcome is the expansion 
of an ‘arbitrary’ odd function in a sine series. Lagrange in 1766 
had constructed by a process of interpolation a finite summation 
formula from which Fourier’s result is obtainable by a leap into 
infinity, but he “abstained from the transition from this sum- 
mation formula to the integration formula given by Fourier.” 21 
Lagrange’s difficulty was that he had a mathematical conscience. 
Physical intuition supplied Fourier’s lack of mathematical 
inhibitions and guided him to the general statement of his 
famous theorem. 

The mathematical physicist’s boldness taught pure mathe- 
maticians several things of the first importance for the future of 
analysis. Purists gradually came to realize that their intuitions 
of ‘arbitrary’ function, real number, and continuity needed 
clarification. P. G. L. Dirichlct’s (1805-1S57, German) defini- 
tion 22 (1837) of a (numerical-valued) function of a (real, numer- 
ical-valued) variable as a table, or correspondence, or correlation, 
between two sets of numbers hinted at a theory' of equivalence of 
point sets. When G. F. B. Ricmann (1826-1866, German) in 
1854 investigated 23 tire representation of a function by' a 
trigonometric (Fourier) scries, he discovered that Cauchy had 
been too restrictive in his definition of an integral, and showed 
that definite integrals as limits of sums exist even when the 


integrand is discontinuous. Later (date uncertain) he invented 
a function, defined by a trigonometric series, which is continuous 
for irrational values of the variable and discontinuous for rational 
values. 24 It was clear that the continuum of real numbers had 
not been thoroughly understood. With our present knowledge 
we see once more what Cantor was the first to perceive, the 
necessity for a theory of sets of points. Cantor’s investigations, 
like Riemann’s, began in Fourier series. 

The demand for clearer understanding of limits, continuity, 
and derivatives was further emphasized in 1874 by the publicity 
given to Weierstrass’ example of a continuous function having no 
derivative or, what is equivalent, of a continuous curve admit- 
ting no tangent at any point. Intuition all but expired. 

Such appear to have been the principal impulses behind the 
creation of the modern continuum. The unforeseen phenomena 
cited, and many others almost equally unexpected but of the 
same general character, seemed to indicate that all the diffi- 
culties were ultimately rooted in the real number system. Urged 
by this conviction, Dedekind, Cantor, and Weierstrass, by 
different methods but with a common aim, returned to the prob- 
lem of Eudoxus and stripped it of its disguised intuitive geom- 
etry. ‘Magnitudes,’ as we have seen, were replaced by ‘numbers,’ 
and geometrical intuitions were driven out to make room for 
those of traditional logic. Nebulous ‘quantities’ persisted in the 
analysis of some. In the numerical epsilons and deltas of rigorous 
Weierstrassian analysis the calculus of the nineteenth century 
attained its classic perfection. The e, 5 technique became part 
of the standard equipment of every working analyst, and an 
advanced course in the calculus toward the end of the century 
usually included the rudiments of Cantor’s theory of sets. 

Finality in 1900 

In the preceding chapter we followed the development of the 
real number system to the close of the nineteenth century, and 
we have just seen that one origin of the modern concept of real 
numbers was analytic necessity. In the retreat of geometric and 
kinematic intuition to the classical logic which validated the 
work of Dedekind, Weierstrass, and Cantor, the calculus 
returned at the close of the nineteenth century to the paradoxes 
of the infinite that had exercised generation after generation of 
logicians from Zeno to Russell. Before further progress was 
possible, a more subtle logical technique had to be developed in 
the twentieth century, and this was forthcoming only when the 


symbolic logic prophesied by Leibniz was extended and refined 
far beyond his utmost imaginings. Thus, after two centuries, the 
calculus returned for new strength and health to one of the 
minds from which it had sprung. What it received will be our 
concern after we have reviewed some of the triumphs of analysis 
in the two centuries following Newton and Leibniz. 

For the moment, we recall the benediction pronounced by 
Henri Poincare (1854—1912, French) at the second international 
congress of mathematicians in 1900. On this historic and some- 
what solemn occasion, Poincare, the outstanding mathematician 
of his epoch and the Lagrange of the nineteenth century, con- 
trasted the roles of intuition and logic in mathematics. In par- 
ticular he reviewed the movement which has just been sketched 
and which, in the late nineteenth century, was called the arith- 
metization of analysis. The comforting assurances of this bold 
master of analysis induced a warm glow of security and pride in 
all who heard him, or who read his memorable address, and who, 
at least temporarily, had forgotten all they knew of mathe- 
matical history. 

Having recalled 53 that mathematicians had once been con- 
tent with the ill-defined and rough images of things mathe- 
matical as they appear to the senses or the imagination, Poin- 
care credited the logicians, for whom he had a dislike sharpening 
occasionally into acid ridicule, with having remedied this un- 
satisfactory state of affairs. Likewise, he continued, for irra- 
tional numbers and “the vague idea of continuity that we owe to 
intuition,” now (1900) resolved into “a complicated system of 
inequalities concerning integers.” By such means, he declared, all 
difficulties concerning limits and infinitesimals had been clarified. 

Today [1900) there remain in analysis only integers and finite or infinite 
systems of integers, inter-related by a net of relations of equality or inequality. 
Mathematics, as we say, has been arithmetined. 

... Is this evolution ended? Have v.-c at last attained absolute rigor? 
At each stage of the evolution our fathers believed that they too had attained 
it. If they deceived themselves, do not we deceive ourselves as they did? 

We believe that sve no longer appeal to intuition in our reasoning. The 
philosophers tell us that this is an illusion . . . 

Now, in analysis today, if sve care to take the pains to be rigorous, there 
are only syllogisms or appeals to the intuition of pure number that could 
possibly deceive us. We may say today (1900) that absolute rigor has been 

Here wc may refer to the last section of the preceding chapter. 
Some of the concrete achievements of analysis in applied mathe- 
matics will be discussed later. 


Rational Arithmetic after Fermat 

We shall conclude our account of number since the seven- 
teenth century with a few typical items from the vast domain of 
the classical theory of numbers. Arithmetic in the tradition of 
Fermat, Euler, Lagrange, Legendre, and Gauss has been con- 
cerned mainly with the rational integers 0, ±1, + 2, . . . . 
Although it has attracted several of the greatest mathematicians 
since the seventeenth century, rational arithmetic has had far 
less influence than its nineteenth-century offshoot, the theory of 
algebraic numbers, on the rest of mathematics. Intensively 
cultivated for its own fascinations by hundreds of mathemati- 
cians of very different tastes, rational arithmetic has developed 
into an ever-growing expanse of loosely coordinated results with 
fewer general methods than any other major division of modern 

From all this heterogeneous miscellany we shall select only 
three topics in which there is some coherence of method and an 
approach to completeness in certain details. The rest is largely 
a wilderness of dislocated facts offering a strange and discon- 
certing contrast to the modernized generality of algebra, geo- 
metry, and analysis. Much of it is hopelessly archaic in both aim 
and results. Rational arithmetic appears to be the one remaining 
major department of mathematics where generalizing a problem 
makes it harder instead of easier. Consequently it has attracted 
fewer merely able young mathematicians than any other. _ _ 

The subject falls naturally into the complementary divisions 
of multiplicative and additive arithmetic. The multiplicative 
theory develops the consequences of unique factorization into 
primes; the additive division is concerned with the composition 



of integers as sums of prescribed types. A capital project in both 
divisions is that of enumeration: how many integers of a speci- 
fied kind satisfy given conditions? For example, how many 
primes are there between given limits? Or in how many ways 
may any integer be represented as the sum of a fixed number of 
positive cubes ? 

A problem in rational arithmetic is said to have been solved 
when a process is described whereby the required information is 
obtainable by a finite number of non-tentative operations. Time 
certainly is not the essence of the contract between the rational 
integers and the human intellect. The problem of resolving a 
number into its prime factors is solvable; yet the finite number 
of operations at present required for a number of a few thousand 
digits might consume more ages than our race is likely to have at 
its disposal. 

The problem of finding the prime factors of a number must 
strike an amateur as a natural one. To say that it has been 
solved in any respect that would satisfy common sense is a 
flattering exaggeration, and the like is true of many other 
arithmetical problems that seem natural to the inexperienced. 
The professional ignores these natural problems in favor of 
others which he or his predecessors have constructed, and for 
which he may hope to find at least partial solutions. Complete 
solutions, even of manufactured problems, are comparatively 
rare; and it would seem that rational arithmetic in the twen- 
tieth century is still relatively in the same position as geometry 
was before Descartes. Compared with what we should like to 
know in each of several directions, such progress as has been 
made is almost negligible. Yet all the resources of algebra and 
analysis have been hurled into the assault on this most elemen- 
tary of all divisions of mathematics. 

Outgrowths of diophantine analysis 

The nature of diophantine analysis has already been 
described in connection with Fermat. Its most extensive out- 
growth, the arithmetical theory of quadratic forms, slowly took 
shape during the eighteenth century, principally in the prodigi- 
ous output of Euler and the more restrained contributions of 
Lagrange and Legendre. Finally, in 1801, with the publication by 
Gauss of his arithmeticae, diophantine analysis 1 
in the sense of Fermat and Euler suffered an eclipse that was to 
last a century, until arithmeticians began to realize that the 


Gaussian theory of quadratic forms does not exhaust the subject 
of indeterminate equations. 

The second great branch of modern arithmetic that sprang 
from diophantine analysis was the theory of congruences. This 
also originated in the Disquisitiones. The suggestiveness of 
Gaussian congruence for modern algebra and the development 
of structural theories was noted in an earlier chapter. 

After Diophantus and Fermat, Euler was the great master of 
indeterminate analysis. But, like nearly all the predecessors of 
Lagrange, Euler contented himself with special sets of integers 
.or of rational numbers satisfying his equations. The only interest 
(such work has had at any time since Lagrange’s discussion of 
x z — Ay 2 = 1 in 1766-9 is in showing that a particular equation, 
or set of equations, with integer coefficients is in fact rationally 
or integrally solvable when the existence of a solution has been 
doubted. Thus a single numerical instance would dispose of the 
doubt (1945) concerning the solvability of a : 4 + y 4 + z 4 = w*, 
xyzzu 0, in integers. In a modern setting, this problem is 
equivalent to determining the number of representations of zero 
in the quartic form x i + y 4 + z 4 — w 4 and, if there are any, 
finding all. Euler (1772) conjectured that there are no solutions. 

The daring of this baseless conjecture typifies the cardinal 
distinction between indeterminate analysis before Lagrange and 
after him. Euler and others in the older tradition did not hesi- 
tate to suggest problems of great difficulty without offering the 
slightest suggestion for a method of attack. And when ingenuity 
furnished special solutions of an equation, the solver dropped the 
matter. Lagrange was the first to impose some common mathe- 
matical morality on diophantine analysis. He refrained from 
facile guessing; and when he did propose a problem, he also 
invented methods for obtaining its solutions. 

The turning point is marked by Fermat’s equation 2 

x 2 — Ay 2 = 1, 

where A is any positive non-square integer, and all integer solu- 
tions x, y are sought. Fermat (1657) asserted that there are an 
infinity of solutions, a fact which Lord Brouncker and J. Wallis 
were unable to prove, although they gave a tentative method, 
improved in 1765 by Euler, for finding solutions. Euler proceeded 
from the conversion of "SjA into a continued fraction. But he 
was unable to prove the existence of a solution with y^O. 
Lagrange (1766-9) supplied the crucial proof, and in 1769-70, 


gave a non-tentative method for obtaining all integer solutions of 
x i _ Ay" 1 = B, where A, B are an y given integers. 

It was noted in earlier chapters that the Pythagoreans 
approximated to quadratic irrationalities by what amounts to 
solving special eases of Fermat’s equation by continued frac- 
tions, and that Brahmagupta in the seventh century gave a ten- 
tative method for solving x- — Ay~ + B in integers. But the 
mathematical distance between such empirical work as this and 
Lagrange’s proofs of necessity and sufficiency is immeasurable; 
and it is fantastic to claim that the Hindu mathematicians 
anticipated Lagrange. There is honor enough for Brahmagupta 
and Bhaskara in having imagined a problem that, centuries 
after they were dead, was to prove of cardinal importance in 
modern arithmetic. But in this they may have been merely 
lucky, for they devoted much time to numerous other problems 
that are essentially trivial. Fermat’s equation and its solution by 
Lagrange are indispensable in the Gaussian theory of binary 
quadratic forms, also in that of algebraic number fields of the 
second degree. Lagrange’s solution tvas the first determination 
of the units in an algebraic number field other than the 
rational. : 

In addition to being haphazard, Euler’s attack on diophan- 
tine equations was absurdly ambitious. If a single equation of the 
second degree in two unknowns proved unexciting, Euler in- 
creased the degree to three or four. If this failed to provide an 
attractive equation, he simultaneously increased the number of 
unknowns. As a last resort, he increased the number of equations 
and exercised his uncqualcd ingenuity on simultaneous systems. 
It is not surprising that he made but little progress toward either 
general methods or general theorems. Nor did any of his hun- 
dreds of successors who equaled or excelled him in ambition, but 
who fell far short of him in ingenuity. 

Advances toward real mathematics began when unambitious 
men like Lagrange and A. M. Legendre (1752-1833, French) 
confined their main efforts to the humble task of systematically 
investigating a single equation of the second degree in not more 
than three unknowns. Their work smoothed the way for Gauss, 
who also set himself a program which, compared to the rank 
opulence of the pre-Lagrangian period, is poverty itself. And 
without the pioneering work of Lagrange and Legendre, it is at 
least doubtful whether even Gauss would have been able to 
compose the Disquisitionrs. 


Arithmetical forms 

The basic technique of the arithmetic of forms originated 
with Lagrange’s theory of binary quadratics, in 1773, four years 
before Gauss was born. To describe it we shall use the standard 
terminology introduced in 1801 by Gauss and modified by later 
arithmeticians. Several of the definitions given next will be useful 
in slightly modified shape when we come to invariance. 

A form in rational arithmetic is a homogeneous polynomial 
P, P(x i, . . . , x n ), in the n indeterminates (or variables) 
Xi, , x n , with integer coefficients. If the degree is m, the 
form is called an 7t-ary m-ic. For n = 2, 3, 4, 5, . . . the forms 
are called binary, ternary, quaternary, quinary, . . . , respec- 
tively. In what follows, ‘form,’ unqualified, shall mean an 
«-ary m- ic. The fundamental concepts are equivalence and 
reduction of forms, and representation by a form. 

The form P{x j, . . . , x n ) is said to contain the form 

Q(4, . ■ ■ , x'„) 

if Q is derived from P by a linear homogeneous substitution 
T :Xi = a i \X l + • • • + ai n x'„{i = 1, . . . , n) with integer co- 
efficients aij whose determinant \a l3 \ is not zero. If \a l3 \ = ±1 
the inverse, T~\ of T, expressing x[, . . . , x n as linear homo- 
geneous functions of Xi, . . . , x n , will have integer coefficients, 
and Q will contain P. When each of two forms thus contains the 
other, the forms are said to be equivalent; the equivalence of P, 
Q is written P ~ Q. It follows readily that this ~ is an instance 
of the abstract equivalence described in connection with Gaus- 
sian congruence. For P is either equivalent or not equivalent to 
Q; P ~ P; and if P ~ Q, then Q ~ P; also, P ~ Q and Q ~ R 
together imply P ~ R. Hence all forms equivalent to a given 
form are equivalent to one another; and therefore all forms may 
be separated into classes with respect to equivalence, two forms 
being put into the same class if and only if they are equivalent. 

The link with diophantine analysis is supplied by the concept 
of representation: an integer r is said to be represented by (or 
in) the form P{x i, . . . , x„) if and only if the equation " 

P(x 1, . * * , ^n) r 

is solvable in integers Xi, , x n . If Xi = s i, . . • ) = /« 

such a solution, (s i, ...,/„) is called a representation of r m P. 
The diophantine problem, as reformulated in 1773 by Lagrange, 


is to decide whether or not a given r is represented in P and, if 
it is, to find all representations. 

It follows at once from the definitions that if r is represented 
by a particular form in a given class, it is represented by every 
form in that class; and that if it is not represented by a particular 
form in the class, it is represented by no form in the class. The 
diophantine problem of finding all integer solutions of 

. . . , .v n ) — t 

is thus reduced to two others: to assign criteria, expressed in 
terms of the given coefficients of P, sufficient to decide whether 
or not r is represented by P; to find all the forms equivalent to P. 
The second of these suggests as a preliminary a third: given the 
coefficients of two forms, to determine whether the forms are 
equivalent and, if so, to transform one into the other. This in 
turn requires the automorphs of a given form, namely, those 
transformations which leave a form unaltered. Once the auto- 
morphs of P and one transformation taking P into Q arc known, 
all such transformations are known. For binary quadratic forms, 
the automorphs are obtained by solving certain of Lagrange’s 
equations x~ — Ay- — B. 

The remaining problem of modernized diophantine analysis, 
that of the reduction of forms, is on a different level. Suppose 
that in each class of forms it is possible to isolate a unique 
form by imposing appropriate conditions on the coefficients of 
all the forms in the class. Then this so-called reduced form, 
being equivalent to ever}' form in the class, may be taken as a 
representative of its entire class in the problems of equivalence 
and representation of numbers. Thus attention may be concen- 
trated on individual forms instead of being dispersed over possi- 
ble infinities of forms in the different classes. Incidentally, the 
problem of determining the number of classes of forms whose 
invariants have any preassigned integer values is suggested. 
Lagrange solved the problem of reduction for binary quadratics 
in 1773; a solution for ternary quadratics was first obtained by 
L. A. Sccbcr (German) in 1831. 

A little trial and a great deal of error will readily convince any 
experimenter that a complete solution of these basic problems 
is not to be anticipated in the immediate future. Nevertheless, 
their mere formulation was a notable achievement. If nothing 
else, they stripped the ancient diophantine analysis of its 
specious simplicity and revealed the nature of its inherent diffi- 


culties. In this respect they are an outstanding example of 
mathematical strategy as practiced by masters. 

These modern, clearly defined problems may prove to be so 
intractable in the general case that they will be abandoned. 
The entire program of a frontal attack on diophantine analysis 
has been questioned. Our successors may be forced to resume the 
classical technique of manufacturing problems which they can 
solve. We recall that diophantine analysis originated in the 
Pythagorean equation x 1 + y 2 = z 2 ; and it is conceivable that the 
generalized problem which sprang from this equation is artificial. 
The Pythagorean equation entered mathematics through geom- 
etry, not through arithmetic. A generation less respectful of 
tradition than ours may succeed in formulating and solving 
problems closer in some as yet unimagined sense to the nature of 
rational arithmetic, whatever that may be. In any event, 
interest in the modernized problems of diophantine analysis 
described above declined rapidly toward the close of the nine- 
teenth century. The problems were simply too hard; and all the 
impressive machinery of modernized algebra and analysis 
succeeded only in making a great clatter which failed to silence 
the insistent questionings of arithmetic. 

By far the major part of all the advances was in the theory 
of quadratic forms. A rough estimate of the amount of work 
done on the several kinds of forms gives eighty per cent in 
quadratic and twenty per cent in all others. Of the work on 
quadratics, about eighty per cent was devoted to binaries, eight 
per cent to ternaries, three per cent to quaternaries, and three 
per cent to n-aries. The remaining six per cent on quadratics was 
accounted for by binaries with coefficients in a few special 
quadratic fields. These statistics suggest that the general pro- 
gram of an arithmetical theory of forms was still largely a hope 
after over a century and a half of industrious exploitation by 
several hundred arithmeticians, including such men as Lagrange, 
Legendre, Gauss, Eisenstein, Dirichlet, Hermite, H. J. S. Smith, 
Minkowski, and Siegel. 

We shall now indicate briefly a few of the outstanding land- 
marks in the theory of forms of low degree. By his general 
treatment of binary quadratics (1773), Lagrange obtained 
incidentally and uniformly many of the special results of his 
predecessors, such as Euler’s theorem (1761) that every prime 
6n + 1 is represented by x 2 + 3y 2 . Lagrange’s principal achieve- 


ment, however, was the introduction of universally applicable 
methods into the theory of binary quadratic forms. 

Legendre in 1798 published his Thcorie des nombres , the first 
treatise devoted exclusively to the higher arithmetic, in which 
Lagrange’s theory was simplified and extended. This work con- 
tains the earliest systematic attack on ternary quadratics. 
Much use was made of the law of quadratic reciprocity, of which 
the first complete proof was published by Gauss in 1801. 

With the Disquisitianes arillimeiicae (1801), the theory of 
binary quadratics crystallized into its classic shape. Systematiz- 
ing and completing details in the work of his predecessors, 
Gauss also added many new ideas of his orvn. Among the innova- 
tions was one which was to prove most unfortunate: Gauss con- 
structed his entire theory on forms ax- -f- 2 bxy -J- cy- with 
a, b, c integers. The even middle coefficient 2b makes the accom- 
panying algebra more elegant but needlessly complicates the 
arithmetic and leads to cumbersome refinements in classifica- 
tion. To an algebraist this may seem a trivial detail. But a 
moment’s reflection will show that as the subject under investi- 
gation is rational arithmetic and not algebra, the insistence that 
the middle coefficient be even is likely to cause unavoidable 

Modern practice for binary quadratics (and to a lesser extent 
for ternaries), following Ivronecker, has returned to the un- 
restricted integer coefficients of Lagrange. Consequently it is 
necessary' to retain two vocabularies and to know which is being 
used in referring to papers on the subject. 

The first man to master the synthetic presentation of Gauss 
was Dirichlct, v'ho in 1863 summarized his personal studies and 
his recasting of the Disquisiliones in his Zahlenthcoric. The 
successive editions (1871, 1879, 1893) of this text 2 and Dirichlet’s 
earlier original contributions made the classical arithmetic of 
Gauss accessible to all without undue labor. A more significant 
advance of Dirichict’s in arithmetic generally will be noted later 
in connection with the analytic theory. 

Up to 1847 the arithmetical theory' of quadratic forms had 
been confined to binaries and ternaries. It might be thought 
that the extension to quadratics in 4, 5, 6, . . . indeterminates 
would be a matter of simple routine, like the passage from three 
dimensions to r. in analytic geometry. Hard experience quickly 
corrects this misapprehension; the difficulties of a detailed 


investigation increase rapidly with the number of indeterminates 
and even necessitate the invention of new principles. 

The first significant departure from the tradition of binaries 
and ternaries was F. M. G. Eisenstein’s (1823-52, German) 
arithmetical determination in 1847 of the number of representa- 
tions of an integer as a sum of six or eight squares. This was 
followed in 1847 and 1850 by an arithmetical determination of 
the number of representations of an integer without square 
factors as a sum of five or seven squares. In all cases only results 
were indicated, with no hint of the methods used. There is no 
doubt, however, that Eisenstein’s procedure was purely arith- 
metical and not analytic. Although all his results have long 
been details in the general theory of n-ary quadratic forms, they 
are of more than casual interest historically, as it was partly 
owing to them that the arithmetical theory was created. 

To trace the development, we must return to the determina- 
tion of the number of representations of an integer as a sum 
of two squares by Legendre in 1798 and, more simply, by Gauss 
in 1801, and to Euler’s unsuccessful struggle for forty years to 
prove that every positive integer is a sum of four integer squares, 
Euler’s failure was the steppingstone to Lagrange’s success in 
1772, and thence to his own a year later. But neither obtained 
the number of representations. Quite unexpectedly the required 
number dropped out as an unsought by-product of an identity 
in elliptic theta constants, which Jacobi encountered in 1828 
while developing the theory of elliptic functions. The like results 
for 2, 6, 8 squares are evident from other formulas in Jacobi’s 
Fundamenta nova of 1829. Those for an odd number of squares, 
lying much deeper, do not follow from similar identities. In 
passing, the problem of three squares was a famous crux in the 
arithmetical theory of quadratic forms until Legendre in 1798 
published the first proof that all positive integers except those 
of the form 4 A (8& + 7) are sums of three integer squares. 

From all this it is clear that Eisenstein made a significant 
advance when he obtained his results for five and seven squares 
arithmetically. Possibly it was this work that moved Gauss to 
assert that “There have been only three epoch-making mathe- 
maticians, Archimedes, Newton, and Eisenstein.” If Gauss ever 
did say this (it is merely attributed to him), it is the most 
astounding statement in the history of mathematics. But as he 
may have said it, and as anything Gauss said about mathematics 
is to be taken seriously, we may briefly examine its tenabihty. 


Like Abel and Galois, Eisenstein was an “inheritor of unful- 
filled renown,” also of poverty and ill-health, and there is no 
guessing what he might have accomplished had he lived. But he 
enjoyed (after a fashion) about two years more of life than Abel, 
and eight more than Galois. His principal achievement outside of 
arithmetic was in elliptic functions, where he partly anticipated 
certain details of the Weierstrassian theory. His own analysis 
halted under the fatal disability of conditional convergence. 4 On 
the other hand, his applications of elliptic functions to the laws 
of cubic and biquadratic reciprocity were among the least ex- 
pected things in arithmetic. Against Gauss’ assertion are the 
facts that for one reference in living mathematics to Eisenstein, 
there are hundreds to Abel and Galois, to say nothing of Rie- 
mann and Dedekind, who were pupils of Gauss, or Eisenstcin’s 
less famous contemporary Rummer. Even in the narrowly 
limited domain of arithmetic, Eisenstein’s influence has been 
slight in comparison with that of Dedekind. In the older form, 
now obsolete, of the theory of higher reciprocity laws, Eisen- 
stein’s work of 1850 was vital; but here again the generative 
concept of prime ideal divisors was Rummer’s. It seems unlikely, 
then, that posterity will revise the almost universal verdict of 
1945, that up till then the three epoch-making mathematicians 
were Archimedes, Newton, and Gauss. 

The principal steps toward a general theory of n-ary quad- 
ratics appear to have been as follows. Perfecting and greatly 
extending the theory for « — 3 as left by Gauss in the Disquisi- 
tiones, Eisenstein in 1847 introduced new principles for the 
classification of ternaries into orders and genera. Hcrmitc in 
1850 simplified the theory of reduction for ternaries, and in 1S51 
devised his general analytic method of continual reduction. The 
theory of ternaries was further developed by Eisenstein in 
1851-2; by H. J. S. Smith (1S26-1S83, Irish) in 1867; by E. 
Selling (German) in 1874, and by many others in the lS50’s-70’s. 
In 1864 and again in 1S67, Smith initiated one form of the 
general theory of r .- ary quadratics from which Eisenstcin’s 
theorems on five and seven squares were easily obtainable. Owing 
partly to the conciseness of the exposition, these and other 
detailed consequences were overlooked, and the problem of five 
squares was proposed by the French Academy for its Grand Prix 
in 1SS2. Brevity in mathematics is sometimes the soul of obscur- 
ity. Smith elaborated the relevant parts of his general theory of 
1S64-7, and shortly after his death shared the prize with H. 


Minkowski (1864-1909, Russian; Germany) then a student of 
eighteen at the beginning of his too brief career. Thus, after an 
unnecessary delay of over a quarter of a century, the general 
arithmetical theory of n-ary quadratics was launched with 
complete eclat. 

Poincare, Minkowski, and others further developed the 
theory in the two succeeding decades. With the exception to be 
noted immediately, little that could be considered basically new 
w r as done till C. L. Siegel (German) in 1935 gave a profound 
reworking of the entire theory. 

The new acquisition, that of the geometry of numbers, was 
created almost entirely by Minkowski, although special in- 
stances 3 of it occur in the early (posthumously published) work 
of Gauss; in a project of Eisenstein’s (1844); in Dirichlet’s 
(1849) asymptotic evaluations of sums of arithmetical functions; 
in the work already cited on the reduction of ternary quadratics; 
and in the semi-geometrical presentation of the theory of elliptic 
modular functions by H. J. S. Smith in 1876, and the similar 
revision of binary quadratics by Poincare in 1880. 

One basic principle is so simple as to appear ridiculous: if 
n -p 1 things are stored in n boxes, and no box is empty, exactly 
one of the boxes must contain two things. The solution of a 
trick problem popular some years ago follows from this principle 
of geometrized arithmetic: state necessary and sufficient condi- 
tions that there shall be at least two human beings in the world 
with the same number of hairs on their heads. 

The first published results in the geometry of numbers appear 
to be Eisenstein’s geometrical proof (1844) of the Gauss lemma 
for the proof of the law of quadratic reciprocity, and his formula 
(1844) for the number of solutions of x 2 -f- y 2 ^ n in integers 
x, y, where n is given. A lattice point being defined as a point 
whose coordinates are integers, the last is equivalent to finding 
the number of lattice points contained by a circle, including its 
circumference, with center at the origin and radius Min- 
kowski developed meager hints like these into a powerful method 
which was applied with conspicuous success by himself and many 
others to difficult questions in the theories of forms, especially 
linear forms with real coefficients, and algebraic numbers. It is 
not necessary, of course, to restate an arithmetical problem 
geometrically; but doing so suggests to those with spacial intui- 
tion in n dimensions analytic processes which they might not 
imagine otherwise. With or without appeal to spacial imagery, 


the type of problem suggested by the geometry of numbers 
inspired much work in the analytic arithmetic developed since 
about 1910 by the schools of E. Landau (1877-1938, German), 
G. Ii. Hardy (18 77-, English), J. E. Littlewood (18SS-, English), 
and S. Ramanujan (1887-1920, Indian) in England. Arithmetic 
thus repaid its heavy debt to analysis by showering some of the 
foremost classical analysts of the twentieth century with an 
abundance of difficult problems. 

The arithmetical theory of forms of degree higher than the 
second was responsible for much less. Eisenstcin initiated (1844) 
the theory of binary cubics, and in so doing came upon the first 
algebraic covariant in history. But he did not exploit his dis- 
covery, although he realized its suggcstivencss. The arithmetic 
of binary cubics was reworked by the British mathematicians 
G. B. Mathews and W. E. H. Berwick in 1912. 

Progress up to 1945 in the arithmetic of forms beyond this 
point, with three exceptions, was inconsiderable. The norm of 
an algebraic integer is the product of all its conjugates; the 
norm equated to unity defines the units of the field concerned. 
Dirichlet (1840) proved the basic theorems for such units, al- 
though there is yet no practicable way of obtaining them even in 
special cases for fields of degree higher than the third. This work 
generalized Lagrange’s on Fermat’s equation. Dirichlct’s prob- 
lem of units and its immediate extension to the representation of 
any number by a general norm are the most immediate generali- 
zation of the theory of binary' quadratics. Today they are a topic 
in algebraic numbers. The origin of this farthest outpost in the 
systematized theory of forms was Lagrange’s remark (1767) that 
the norm of a general algebraic number repeats under multiplica- 
tion, and this in turn can be traced back to Fibonacci’s identity. 
It may interest some to know that Diricblet’s inspiration came 
to him in church while he was listening to the music on an Easter 

The second exception to the general rule of sterility beyond 
real quadratics was Hcrmitc’s introduction in 1854 and 1857 of 
the forms since known by his name. In the binary case, a 
Hcrmitian form is of the type ax: v' -f- bxy’ -f b’x'y -f- cry', where 
a, c are real constants, b , V conjugate imaginary constants, and 
the variables in the pairs a\ x' and y, y f arc conjugate imaginarics, 
so that the entire form is real and hence capable of representing 
real numbers. From Ilermite’s arithmetical theory' of these 
forms in two or more variables evolved the extensive theory of 


Hermitian forms and matrices, which after 1925 became familiar 
to physicists through the revised quantum theory. Hermite also 
( 1849 ) initiated the closely related arithmetical theory of bilinear 
forms, 6 thus starting much algebra that is now standard in a 
college course, including parts of the theory of matrices and 
elementary divisors. The last originated explicitly in H. J. S. 
Smith’s discussion of systems of linear diophantine equations 
and congruences (1861), and was developed independently by 
Wcierstrass and by G. Frobenius (1849-1917, German) in the 
1870’s-80’s. The point of historical interest here is that all these 
extremely useful techniques of modern algebra, which after 1925 
became commonplaces in mathematical physics, evolved from 
quite useless problems in the theory of numbers. 

The third and last exception to general sterility connects 
the arithmetic of forms with that other major outgrowth of 
ancient diophantine analysis, the Gaussian concept of con- 
gruence. Dickson in 1907 began the congruencial theory of forms, 
in which the coefficients of the forms are either natural integers 
reduced modulo p, p prime, or elements of a Galois field. The 
linear transformations in the theory, corresponding to those in 
the classical problem of equivalence, were similarly reduced, and 
hence modular invariants and covariants were definable. By 1923 
the theory was practically worked out, except for two central 
difficulties, by Dickson and his pupils. Simplified derivations for 
some of the results were given (1926) by E. Noether by an 
application of her methods in abstract algebra. 

Before passing on to congruences, we note an outstanding 
advance in the older tradition of diophantine analysis. If 

/(z) = a n z n + a n _iz n_1 + • • • + a# + a 0 

is an irreducible polynomial of degree n ^ 3 with integer coeffi- 
cients, and if 

H(x, y ) s a n x n + tfn-i* n-3 y + * * * + dixy"” 1 + a 0 y n 

is the corresponding homogeneous polynomial, then H(x, y) = 
where c is an integer, has either no solution or only a finite num- 
ber of solutions in integers x, y. This is the capital theorem (1909) 
of A. Thue (Scandinavian). A generalization 7 was given by Thue 
himself, and another (1921) by Siegel. After all that has been 
said aboup the paucity of general methods and the plethora o 
fragment.* ^ results in diophantine analysis, Thue s theorem 
speaks forltself. It was proved by elementary methods. 


The theory of congruences 

The subject of congruences in rational arithmetic is usuallv 
assigned to the multiplicative division, although it is concerned 
chiefly with the detailed investigation of one highly specialized 
type of diophantine equation. a T x~ -f - • • -f a ; .v tCo= my, 
in which*, y are theindeterminates. and the coefficients a n , . . . , 
a u o D , vi are given constant integers, with a n ^ 0, m 0. The 
essential point is that one of the indeterminates, y, occurs only 
to the first degree. The cases v: = ±1, being of no interest, are 
excluded. The equation rewritten as a congruence is 

o„x Tt -f- • * * -r oi.x -f Co s 0 mod m ; 

m is called the modulus; ?: is the degree; and solving the dio- 
phantine equation is equivalent to finding all integer values of 
x, called the roots of the congruence, that make the polynomial 
on the left a multiple of m. If x = c is a solution, so also is 
c -f km, where k is any integer. Since c 4- km s c mod in, it 
suffices to find all solutions whose absolute values do not exceed 
\rr\j2. These arc said to be incongruent modulo m. The statement 
of the generalized problem for one congruence in several indeter- 
minates, or for a simultaneous system of such, is immediate. We 
refer here to what has already been said about congruences in 
connection with algebraic structure. 

Gauss does not disclose 5 what led him to this cardinal con- 
cept of modern arithmetic. But its systematic use in the earlier 
sections of the Disquisiiiones enabled him to unify and extend 
important theorems of Fermat, Wilson, Euler, Lagrange, and 
Legendre on arithmetical divisibility and, in the famous seventh 
(concluding) section, to give a reasonably complete theory of 
binomial equations in algebra. A few of the older results on divisi- 
bility maybe restated in the language of congruences to illustrate 
the general ideas. 

If p is prime, ar* -1 — 1=0 mod p, has exactly p — 1 incon- 
gruent roots. This is Fermat’s theorem, proved essentially by 
the implied method of congruences by Lagrange (1771), who 
showed also (176S) that a congruence of degree r. has not more 
than v. incongruent roots for a prime modulus. The theory of 
residues of powers, originating with Euler in 1769, is concerned 
with the general binomial congruence x n - enfl mod m. It has 
numerous applications in algebra, particularly in the theory of 
equations and in finite groups. One crucial unsolved problem 


may be noted. If p is prime, r any number not divisible by p and 
if p — 1 is the least value of n for which r n — 1=0 mod p, 
r is called a primitive root of p. A prime p always has exactly 
4>{p — 1) primitive roots, where p{n) is Euler’s function denoting 
the number of positive integers not greater than n and prime 
to n. The problem is to devise a practicable, non-tentative 
method for finding at least one primitive root of any given prime. 
Between Euler’s initiation of the subject in 1769 and 1919 — a 
century and a half — 232 lengthy articles and short notes were 
published on binomial congruences. None made any substantial 
progress toward a solution of this crucial problem. 

Another outgrowth of Euler’s power-residues is the entire 
theory of reciprocity laws, already noted in connection with 
modern algebra. Yet another is the extensive theory of functions 
defined for integer values of their variables, all of which evolved 
from the theorem of Gauss that 2<£(d) — n i where the summa- 
tion extends to all divisors d of the fixed integer n, and p is 
Euler’s function. 

Practically every arithmetician of note and a host of humbler 
workers have contributed to the theory of congruences since 
Gauss started the subject in 1801. For all their efforts, two cen- 
tral problems of the theory defy solution: to assign criteria on the 
given coefficients of a system (one or more) of congruences to 
decide whether or not the system is solvable and, if it is, to find 
all its incongruent solutions non-tentatively. For a single con- 
gruence of the first degree in any number of unknowns, the 
problem is more completely solved than are most of the solved 
problems in arithmetic, and likewise for a simultaneous system 
of such congruences, the solution having been given (1861) by 
H. J. S. Smith. The higher reciprocity laws represent the farthest 
advance in the case of binomial congruences; their complexity 
may hint that the general problem is intractable by methods 
known up to 1945. It must be remembered that these problems 
and others like them in classical arithmetic have not been the 
easy sport of mediocre men; some of the most powerful mathe- 

maticians in history have wrestled with them. 

Congruences were responsible for one theory of far more than 
merely arithmetical interest. The notation for a congruence 
suggests the introduction of appropriate ‘imaginaries to supply 
the congruence with roots equal in number to the degree of the 
congruence when there is a deficiency of real roots. As in t e 
corresponding algebraic problem, it is not obvious that xmagi 


narien can be introduced consistently. That they can, was first 
proved in 1830 by Galois, who invented the required ‘numbers,’ 
since called Galois imaginaries, for the solution of any irreducible 
congruence Fix) — 0 mod p, where p is prime. He thus obtained 
a generalization of Fermat’s theorem, and laid the foundation 
of the theory of finite fields. As remarked by Dickson, 5 “Galois’s 
introduction of imaginary" roots of congruences has not only led 
to an important extension of the theory of numbers, but has 
given rise to wide generalizations of theorems which had been 
obtained in subjects like linear congruence groups by' applying 
the ordinary theory of numbers.” Galois was eighteen when he 
invented his imaginaries. 

We pass on to the third and last division of rational arith- 
metic which we shall consider. Here great progress has been 
made, most of it since 1895. 

Applications of analysis 

Since the time of Euler, analysis has been applied incidentally 
to rational arithmetic; but it was only in 1839, with Diri- 
chlct’s Recherches sur diverses applications dc 1' analyse infinitesi- 
ntale a la iheorie dcs nombres , that limiting processes entered 
organically into the theory of numbers. Before Diricblet, such 
analysis as was used remained in the background , y ‘ arithmetical 
results being obtained by' the device of comparing coefficients in 
two or more expansions of a given function by different algo- 
rithms. This technique originated with Euler in his work of 1 7*1 8 
in the theory of partitions, a subject which he initiated in 1741. 

After Dirichlct, the next organic uses of analysis were 
Hermite’s (1851) in his method of continual reduction, and 
Ricmann’s (1859) in the distribution of primes. But it was not 
until the twentieth century' that modern analysis was sys- 
tematically ' applied to additive arithmetic. Simultaneously' there 
was an unprecedented advance in the application of analysis to 
the multiplicative division. We shall presently' describe only 
enough to illustrate the radical difference between the old and 
the new. With the exception of a very' few outstanding acquisi- 
tions like Thue’s theorem, it seems probable that the early 
twentieth century will be remembered in the future history of 
rational arithmetic chiefly for its achievements in the analytic 

There remains, however, what some consider a desideratum: 
to obtain those results of the analytic theory' which do not 


involve a limiting process in their statement without an appeal 
to continuity. Thus Dirichlet proved analytically that there are 
an infinity of primes of the form an + b, where a, b are constant 
relatively prime positive integers and n runs through all posi- 
tive integers. Attempts by Emmy Noether and others in the 
1930’s to obtain this non-analytic theorem without the use of 
analysis failed. On the other hand, all of Euler’s and Jacobi’s 
theorems on partitions, and all of the non-analytic theorems on 
numbers of representations in certain quadratic forms, first 
obtained analytically by Jacobi and others, have been proved 
without analysis. The reasons for failure in one instance and 
success in another superficially indistinguishable from it are 
not understood. It will be convenient to call demonstrably 
avoidable analysis inessential; and to speak of analysis which has 
not yet been proved inessential, or which leads to final results 
implying a use of continuity, as essential. Kronecker would 
probably not have admitted essential analysis into arithmetic, 
and might even have declared that its products are as inexistent 
as irrationals. 

A classic example of the use of analysis later seen to be 
inessential occurs in many of the applications of elliptic and 
modular functions to the Gaussian theory of binary quadratic 
forms. In the hands of Kronecker, Hermite, and a score of less 
notable mathematicians, a close connection between binary 
quadratics and the theory of complex multiplication of elliptic 
functions was developed after 1860 into an extensive department 
of arithmetic. One detail of all this intricate theory exemplifies 
the analytic peculiarities. From a passage in the Disquisitiones 
it appears that as early as 1801 Gauss had effected the difficult 
determination of the number of classes of binary quadratics 
having a given determinant. The first published determination 
was Dirichlet’s of 1839, in which analysis was essential. (An 
outstanding desideratum is a ‘finite’ proof of Dirichlet’s results.) 
For forms of a negative determinant, Kronecker in 1860 found 
several remarkable formulas whereby the number of classes can 
be calculated recursively without analysis. These formulas ap- 
peared as by-products of Kronecker’s investigations in elliptic 
functions, and were the heralds of several hundred by later 
writers, many of whom used elliptic modular functions to obtain 
their results. We shall not pursue this matter further here, as a 
detailed account belongs to the specialized theory of numbers 
and we can attend only to matters of more than local significance. 


The point of interest here is that whereas analysis was 
essential in Dirichlct’s derivation of the class-number, and might 
therefore have been reasonably expected to appear essentially 
in the deduction of the recurrence relations, it actually proved 
to be inessential for the latter- Arithmeticians who insist (there 
arc such) that a method containing analysis essentially 11 belongs 
to analysis and not to arithmetic would claim that Kroneckcr’s 
formulas, not Dirichlct’s, are the arithmetical solution of the 
class-number problem. 

There is more than a pedantic difference between the two 
opinions, at least historically. Experience has shown that the 
search for proofs and theorems independent of essential analysis 
frequently turns up unexpected simplicities and reveals new 
arithmetical phenomena. Gauss emphasized the desirability of 
multiplying proofs in arithmetic with a view to making the 
abstruse clear. However, arithmetic is sufficiently broad and 
difficult to permit all types of workers to follow their own 
inclinations. Beginning about 1917, the general trend was toward 
essential analysis. 

The theory of partitions illustrates the historical disconti- 
nuity between inessential and essential analysis. If P(n) denotes 
the total number of ways the positive integer n is obtainable as 
a sum of positive integers, it is obvious, as noted by Euler in 
1748, that P(n) is the coefficient of x n in the expansion of 

[11(1 - * r )] 1 into a power series in x. With unsurpassed 

manipulative skill, Euler derived numerous identities between 
this infinite product and others suggested by problems in parti- 
tions, thus anticipating many formulas in elliptic theta constants 
dcduciblc from Jacobi’s presentation (1828-9) of elliptic func- 
tions. An extensive literature sprang from these discoveries. 
Much of it is algebraic, and in none is analysis essential. Indeed, 
Sylvester, no analyst, desiring to understand the subject and 
being too impatient to master elliptic functions, developed a 
hint thrown out by N. M. Ferrers (1S29-190S, English) in 1853 
into a graphical theory in "which some of the properties of parti- 
tions can be inferred from point lattices. But although the 
pictorial representation may have enabled Sylvester and some 
others to avoid inessential analysis in their thinking, it added 
nothing new to the theory of partitions. 

Among other results of the Euler-Jacobi tradition were 
numerous elegant formulas whereby P(n) and other partition 


functions could be calculated recurrently. Each afforded a com- 
plete arithmetical solution of the problem of computing the 
function of n concerned for any value of n. That is, all were use- 
less as aids to practical computation for any but inconsiderably 
small numbers. 

The break came in 191 7, when G. H. Hardy (1877- English) 
and S. Ramanujan (1887-1920, Indian) applied their new 
analytic methods to the derivation of an asymptotic formula for 
P(n), which put the practical evaluation of P(n) into touch with 
common sense. Before this, it had required a month’s labor by a 
leading expert in the older methods to compute P(200); only six 
terms of the asymptotic formula gave 

P(200) = 3,972,999,029,388 

with an error of .004. This detail typifies the computational 
superiority of formulas depending on essential analysis over the 
corresponding exact theorems preferred by pure arithmeticians. 
The analysis used is applicable to many other numerical func- 
tions appearing as coefficients in expansions of certain functions 
of a complex variable not continuable beyond the unit circle. 
This work of 1917, like most of the essential analysis in modem 
rational arithmetic, was of more than local interest in its own 
domain. It instigated a greatly increased activity in refined 
classical analysis and in the modem theory of inequalities. 

One of the most famous problems in all arithmetic, that 
of the distribution of primes, yielded to analysis only in 1896, 
when J. Hadamard 11 (1865—, French) and C. J. de la Vallee- 
Poussin (Belgian) proved independently that the number N(x) 
of primes sS x is asymptotically equal to ar/log x , that is, the 
limit of N{x) - [.r/log at ] -1 as x tends to infinity is 1. This is 
usually called the prime number theorem. The relevant history 
would fill a book. 13 Legendre, Gauss, and others had proposed 
formulas inferred from actual counts of primes; but it cannot be 
said that any of these tentative efforts materially furthered the 
final success. The great Russian mathematician, P. Tchebycheff 
(1821-1894) in 1830-1 made the first considerable advance since 
Euclid in the theory of primes; but much sharper analysis was 
needed than any available until the last decade of the nineteenth 
century. What appears likely to remain for some time the best 
proof of the theorem is Landau’s reformulation (1932) of that of 
N. Wiener (1894—, U.S.A.), who deduced the result almost as a 
corollary from his work on Tauberian theorems. The latter, so 



named by Hard)* after the German analyst Tauber, evolved from 
the converse of Abel’s theorem on convergent power series. 

Even the briefest notice of the theory of primes must men- 
tion the famous conjecture known as Riemann’s hypothesis, 
which is to classical analysis what Fermat’s last theorem is to 
arithmetic. Euler (1737) noted the formula Stt - ’ = 11(1 — p ~- *)~ 1 , 
the sum extending to all positive integers n, and the product to 
all positive primes p. The necessary conditions of convergence 
hold for complex values of s with real part >1. Considering 
2?z - * as a function f(r) of the complex variable s, Riemann 
(1859) proved that f(r) satisfies a functional equation involving 
£(j), f(l — s), and the gamma function of s. He was thus led 
to the theorem that all the zeros of f (r), except those at s = —2, 
—4, —6, . . . , lie in the strip of the r-planc (Argand diagram 
for s) for which 0 ^ cr < 1, where a is the real part of s. His 
theorems would be even more interesting than they are if all the 
zeros in the strip should lie on the line cr = 4-. Riemann conjec- 
tured 14 that this is so. Attempts to prove or disprove this con- 
jecture have generated a vast and intricate department of 
analysis, especially since Hardy proved (1914) that f(s) has an 
infinity of zeros on cr = h. Although the question was still open 
in 1945, scores of profound papers bristling with thorny analysis 
had enriched the literature of analytic arithmetic for almost a 
third of a century, some of them based, however, on the sup- 
position that Riemann’s conjecture is true. 

This bold technique of inference from doubtful conjectures 
was something of a new departure in arithmetic, where the 
tradition of Euclid, Lagrange, and Gauss had stickled for proof 
or nothing. The rationalized justification for the novel proce- 
dure, were any needed, was the unrealized hope that by trans- 
forming a dubious hypothesis into something new and strange, 
an accessible equivalent would sometime, somehow, drop out. 
Still on the assumption that Riemann's hypothesis and other 
unproved conjectures of a similar character are true, numerous 
profound theorems on the representation of numbers as sums 
of primes, or in other interesting forms, were skillfully deduced 
by some of the most refined analysis of the twentieth centur}'. 
Should any of these boldly conceived but unborn theorems ever 
materialize, they will be among the most remarkable in 

Adhering more strictly to the Euclidean tradition of proof 
before prophecy, the Russian mathematician I. M. Vinogradov, 


beginning about 1924, developed new methods in the analytic 
theory of numbers, and in 1937 apparently came within a 
reasonable distance of proving another famous guess concerning 
primes: every even number >2 is a sum of two primes. C. Gold- 
bach (1690-1764, Russia) in 1742 confided this conjecture to 
Euler who, while believing it to be true, confessed his inability 
to prove it. Anyone who has inspected the analysis by which 
Vinogradov proved (1937) that every odd number beyond a 
certain point is a sum of three odd primes will sympathize with 
Euler. The best previous result was that of L. Schnirelmann 
(Russian), who proved (1931) that there is a constant n such 
that every integer >0 is a sum of n or fewer primes. But the 
method of proof, according to Landau, was incapable of further 
refinement. Vinogradov’s theorem was conjecturally derived in 
1923 by Hardy and Littlewood from an unproved mate of 
Riemann’s hypothesis. 

From 1896 till 1940 a major part of analytic arithmetic 
originated in the theory of rational primes. Some of this was 
extended to algebraic numbers, as when Landau (1903) obtained 
the prime ideal theorem corresponding to, and including, the 
prime number theorem. Here the necessary analysis proceeds 
from Dedekind’s generalization (1877) of Riemann’s £(s) to 
algebraic number fields. Another, less extensive, research gen- 
eralized Dirichlet’s work on the class-number of binary quad- 
ratics, being concerned with the number, proved to be finite, 
of distinct classes of (integral) ideals in an algebraic number 
field. The explicit determination of this number in an approach- 
able form is one of the unresolved cruxes in arithmetic. All of 
this work belongs to the multiplicative division. 

Equally prolific of new analysis and far-reaching theorems 
in arithmetic was the advance beginning in 1909 with Hilbert’s 
solution of Waring’s problem. The English algebraist E. Waring 
(1734— 1798) emitted the conjecture (1770) that every integer 
n > 0 is the sum of a fixed least number g(s) of sth powers of 
integers ^0. For s = 2, this is the result proved by Lagrange 
and Euler that every positive integer >0 is a sum of four 
integer squares ^0. Since no integer 4 A (8& + 1) is a sum of 
three squares, it follows that g( 2) = 4; and it is known that 
g(3) = 9. Waring himself proved no single case of his problem; 
nor did he offer any suggestion for its solution. For all. that he 
or anyone else in the eighteenth century knew, g(s) might not 


It so happens, however, that Waring’s guess was one of those 
few in the theory of numbers that have started epochs in 
arithmetic. Little of any significance issued from Waring’s 
conjecture until about a century and a half after it had been 
made on only scanty numerical evidence. The theorem itself 
might be guessed after an hour’s figuring. For example, g{ 4) 
may be 19 as stated by Waring, a result which had not been 
proved as late as 1945. 

It used to be imagined by romanticists that Waring and other 
rash guessers in arithmetic knew mysterious methods, now lost, 
of extraordinary power. There is no evidence that they did not. 
But professionals who appreciate the inherent difficulties of 
arithmetic believe that the lost methods, with the possible excep- 
tion of Fermat’s, are mythical. Gauss, for example, wdicn urged 
in 1818 to compete for a prize offered by the French Academy 
for a proof or disproof of Fermat’s last theorem, expressed him- 
self quite forcibly on the undesirability of facile guessing in 
arithmetic. Including Fermat’s theorem in his remarks, Gauss 
declared that he himself could manufacture any number of 
such conjectures which neither he nor anyone else could 

All questions of mathematical ethics aside, it is at least 
possible that stating difficult problems with no hint of a method 
for attacking them is more detrimental than advantageous to 
the progress of arithmetic. Unless we arc eager to believe that 
certain individuals are divinely inspired and can foresee what 
course mathematics should follow to accord with the inscrutable 
verities of Plato’s Eternal Geometer, we may suspect that base- 
less guessing is likely to deflect talented originality into artificial 
channels. Waring was an accomplished algebraist, but there is 
no evidence that he was inspired; and it seems like nothing but 
blind luck that his easy guess led to anything more profound 
than trivialities. That it did finally prove extremely stimulating 
may appear in the long run disastrously unfortunate. For there 
is little doubt that Waring's delayed success was largely responsi- 
ble for the return about 1920 to the prc-Lagrangian tactic of 
deduction based on conjectures. Of course Lagrange and Gauss 
may have been mistaken or merely pedantic in their rejection 
of published guessing as a stimulus to progress, and the ninc- 
tccnth-ccntury caution may have been excessive. If so, the early 
twentieth century will doubtless be long remembered as the 
dawn of a new era in arithmetic. 


Hilbert’s proof (1909) of Waring’s conjecture established the 
existence of g(s) for every s, but did not determine its numerical 
value for any s. The curiously ingenious proof, shortly simplified 
by several mathematicians, depended on an identity in 25-fold 
multiple integrals, and like much of its author’s mathematics 
aimed only at existence without construction. Its historical im- 
portance is less that it was the first solution of an outstanding 
problem than that it incited analysts to find at least a bound to 
the numerical value of g(s) for any given s. It was the latter 
problem and the cognate one to be described presently that 
were largely responsible for the explosive outburst of analytic 
arithmetic in the 1920’s-30’s. As already implied, this work 
marks an epoch in the theory of numbers. 

Hardy and Littlewood in 1920-8 invented the analytic 
method for Waring’s problem which was to remain the stand- 
ard till Vinogradov, having started in 1924 from methods similar 
to those of the English arithmeticians, developed his own more 
penetrating technique in the 1930’s. The problem affiliated with 
g(j-) is that of finding G(r), defined as the least integer n such 
that every positive integer beyond a certain finite value is the 
sum of n 5th powers of integers ^0. Thus the best value of g(4) 
up to 1933 was g(4) g 35, in contrast with Hardy and Little- 
wood’s G( 4) 19, while it was shown (1936) that G(4) is either 
16 or 17. For s > 6, Vinogradov’s improved methods gave (1936) 
much smaller values of G(s ) than those obtained previously. 

Although the pioneering methods were thus superseded, their 
influence on the development of asymptotic analysis remained 
incalculable. Utilizing results of Vinogradov’s, Dickson and 
S. S. Pillai (1902-, Indian) in 1936 proved independently an 
explicit formula for g(s), valid for all s > 6, except possibly for 
certain doubtful cases. 15 It is gratifying to report that since the 
first edition (1940) of this book, these doubtful cases have been 
disposed of (1943) by I. M. Niven (1915-, U.S.A.). 

Thus, after 169 years Waring’s guess was finally proved. 
In addition to instigating a vast amount of acute analysis, the 
problem had suggested numerous others solvable by similar 
methods, such as the representability of all, or ‘almost all, 
positive integers as sums of polynomials taking integer values for 
integer variables, or as sums of squares and primes. As this is 
written, there is no sign of an abatement in the output of analyt- 
ic arithmetic. Two of the leading experts in the subject, 
Ramanujan and Landau, died before their time. The others 


more directly responsible for the creation of new methods were 
still active in 1945, and a crowd of younger men was coming on. 

Another isolated result solidified by modern analytic methods 
may be cited for its curious implications regarding the sup- 
pressed work of the initiator of the modern era in arithmetic. 
C. L. Siegel (1 896 — , Germany, U.S.A.) in 1944 gave the first 
proof of a statement (1801) by Gauss concerning a certain 
asymptotic mean value in the theory of the binary quadratic 
class number. As the relevant formula could hardly have been 
inferred from numerical examples, it would be interesting to 
know how Gauss satisfied himself of its correctness. In any 
event, it is indicative of the lawless difficulties of the theory of 
numbers that a result stated by Gauss should have stood in the 
classic literature for 143 years without proof. 

The 1920’s— 1 930’s witnessed the beginning of an era in arith- 
metic comparable to that inaugurated by Gauss in 1801. 
Analysis, the mathematics of continuity, had at last breached 
outstanding problems in the domain of the discrete. That explicit 
integer values for numerical functions like g(j), G(r) should 
be obtainable by analysis would have seemed miraculous to 
the arithmeticians of the nineteenth century. The like holds 
for the modern work in the theory of primes, and in other parts 
of multiplicative arithmetic. It is therefore not true, at least 
in the theory of numbers, that all the great mathematicians died 
before 1913. 

In this chapter, as in others, literally hundreds of worthy 
names have been passed over without mention, and likewise for 
dozens of extensive developments to which scores of workers in 
the past two centuries have devoted their lives. The topics 
described arc, however, a fair sample of some of the best that 
has been done in rational arithmetic since Fermat. 


Contributions from Geometry 

With a literature much vaster than those of algebra and 
arithmetic combined, and at least as extensive as that of 
analysis, geometry is a richer treasure house of more interesting 
and half-forgotten things, which a hurried generation has no 
leisure to enjoy, than any other division of mathematics. Con- 
tinually changing ideals and objectives in the development of 
geometry since the seventeenth century have made it impossible 
for students and working mathematicians to be aware of hun- 
dreds of theorems, and even extensive theories, that the geom- 
eters of the late nineteenth century prized as objects of rare 

On a rather humble level, for example, it was held by compe- 
tent geometers in 1940 to be a sheer waste of effort for a student 
contemplating a career in geometry, or in any division of living 
science or mathematics, even to glance at the so-called modern 
geometry of the triangle and the circle, created largely since 
1870. Yet it has been said, no doubt justly, that almost any 
theorem of this intricate and minutely detailed subject would 
have delighted the ancient Greeks. And that, precisely, is the 
point. All the classic Greek geometers were buried or cremated 
two thousand years ago. Geometry in the meantime has ad- 
vanced. By 1900 at the latest, special theorems in Euclidean 
geometry were no longer even a tertiary objective of creative 
geometers, no matter how beautiful or how interesting they 
might appear to their authors. 

This does not imply that such theorems were valueless to 
those who could appreciate them; they preserved more than one 
isolated teacher from premature fossilization. They may also 
have irritated some who later became skilled geometers into 




finding out what modern geometry is about. On the other 
hand, many a working mathematician of the 1930’s looked back 
with something akin to rage in his heart on the months or years 
squandered on this sort of geometry, or on the interminable 
properties of conics, at the very time of his life when his capacity 
for learning was greatest and when he might have been master- 
ing some living mathematics. 

In defense of this waste, if such it were, it was argued that 
English schoolboys stili took a keen pleasure in these intriguing 
puzzles of their forefathers. No doubt they did. But the further 
claim that such a training made first-rate geometers is con- 
tradicted by the evidence. The attempted justification on the 
grounds of mental discipline may be left to the psychologists. 
In any event, it seemed slightly fatuous to impart discipline 
through outmoded fashions when so much of equal difficulty 
and vital necessity had to be mastered if one were to think 
geometrically in the manner demanded by a continually advanc- 
ing science and mathematics. The foregoing opinions, it may be 
stated, were those of three of England’s foremost mathemati- 
cians in the 1930’s, all of whom had made high marks in this 
prehistoric sort of geometry in competitive examinations for 
English university scholarships. 

However, even in the most elementary geometry an original 
and ingenious mind may occasionally think of something to do 
rather unlike what is already classical. The more orthodox 
Greek geometers, we saw, limited themselves to a straightedge 
and compass in their permissible constructions. Why not dis- 
pense with one or other of these traditional implements? It 
occurred to G. Mohr (Danish) in the seventeenth century to 
sec what constructions could be performed with a compass 
alone, and L. Maschcroni, (1750-1800, Italian) actually %vrote 
a book on the geometry of the compass. Napoleon Bonaparte 
is said to have been highly elated by his solution of a simple 
problem in Maschcroni’s geometry. Others hobbled themselves 
by using a straightedge alone, or this with one given point in 
the plane, and so on. Finally it occurred to E. Lemoinc (1840— 
1912, French), to attempt to assign a measure of the com- 
plicatcdness of a geometrical construction. He presented an 
account of his proposals (18SS-18S9, 1892-1S93) at the Inter- 
national Mathematical Congress held in connection with the 
Chicago World Fair of 1893. He succeeded in defining the 
simplicity of a construction in terms of five operations of clc- 


mentar y geometry, such as placing one point of a compass on a 
given point; the simplicity -was the total number of times these 
operations were used. It was thus possible to assign marks to 
different constructions for the same figure, but a method 
for finding the construction with an irreducible number of marks 
seemed to be lacking. Once more it was demonstrated that the 
only royal road to elementary geometry is ingenuity. Another 
type of problem the beginner meets may have suggested the 
hotly controversial and partly discredited theory of enumerative 
geometry: how many lines, circles, etc., satisfy a prescribed set 
of conditions? Or, to take an instance from the 1940’s, what 
practical use, if any, can be made of Pascal’s theorem in conics? 
Almost exactly 300 years separate Pascal’s discovery of the 
theorem and its application by aeronautical engineers to lofting; 
naval architects might well have used it earlier. At a slightly 
more advanced stage, almost anyone can invent his own peculiar 
system of coordinates and proceed to elaborate the geometry 
it suggests. Many have. 

Rising to a considerably higher level of difficulty, we may 
instance what the physicist Maxwell called “Solomon’s seal in 
space of three dimensions,” the twenty-seven real or imaginary 
straight lines which lie wholly on the general cubic surface, 
and the forty-five triple tangent planes to the surface, all so 
curiously related to the twenty-eight bitangents of the general 
plane quartic curve. If ever there was a fascinating snarl of 
interlaced theories, Solomon’s seal is one. Synthetic and analytic 
geometry, the Galois theory of equations, the trisection of 
hyperelliptic functions, the algebra of invariants and covariants, 
geometric-algebraic algorithms specially devised to render the 
tangled configurations of Solomon’s seal more intuitive, the 
theory of finite groups — all w^ere applied during the second half 
of the nineteenth century by scores of geometers who sought to 
break the seal. 

Some of the most ingenious geometers and algebraists in 
history returned again and again to this highly special topic. 
The result of their labors is a theory even richer and more 
elaborately developed than Klein’s (1884) of the icosahedron. 
Yet it was said by competent geometers in 1945 that a serious 
student need never have heard of the twenty-seven lines, the 
forty-five triple tangent planes, and the twenty-eight bitangents 
in order to be an accomplished and productive geometer; and 
it was a fact that few in the younger generation of creative 



geometers had more than a hazy notion that such a thing as 
tiie Solomon’s seal of the nineteenth century ever existed. 

Those rvho could recall from personal experience the last 
glow of living appreciation that lighted this obsolescent master- 
piece of geometry and others in the same fading tradition looked 
back with regret on the dying past, and wished that mathe- 
matical progress were not always so ruthless as it is. They also 
sympathized with those who still found the modern geometry 
of the triangle and the circle worth cultivating. For the differ- 
ence between the geometry of the twenty-seven lines and that of, 
say, Tucker, Lemoine, and Brocard circles, is one of degree, 
not of kind. The geometers of the twentieth century long since 
piously removed all these treasures to the museum of geometry, 
where the dust of history quickly dimmed their luster. 

For those who may be interested in the unstable esthetics 
rather than the vitality of geometry, we cite a concise modern 
account 1 (exclusive of the connection with hyperclliptic func- 
tions) of Solomon’s seal. The twenty-seven lines were discovered 
in 1S49 by Cayley and G. Salmon 2 (1S19-1904, Ireland); the 
application of transcendental methods originated in Jordan’s 
work (1S69-70) on groups and algebraic equations. Finally, 
in the 1870’s L. Cremona (1830-1903), founder of the Italian 
school of geometers, observed a simple connection between 
the twenty-one distinct straight lines which lie on a cubic 
surface with a node and the ‘cat’s cradle’ configuration of 
fifteen straight lines obtained by joining six points on a conic 
in all possible ways. The ‘mystic hexagram’ of Pascal and its 
dual (1806) in C. J. Brianchon’s (1783-1864, French) theorem 
were thus related to Solomon’s seal; and the seventeenth 
century met the nineteenth in the simple, uniform deduc- 
tion of the geometry of the plane configuration from that of 
a corresponding configuration in space by the method of 

The technique here had an element of generality that was to 
prove extremely powerful in the discovery and proof of cor- 
related theorems by projection from space of a given number of 
dimensions onto a space of lower dimensions. Before Cremona 
applied this technique to the complete Pascal hexagon, his 
countryman G. Veronese had investigated the Pascal configura- 
tion at great length by the methods of plane geometry, as had 
also several others, including Steiner, Cayley, Salmon, and 
Kirkman. All of these men were geometers of great talent; 


Cremona’s flash of intuition illuminated the massed details of 
all his predecessors and disclosed their simple connections. 

That enthusiasm for this highly polished masterwork of 
classical geometry is by no means extinct is evident from the 
appearance as late as 1942 of an exhaustive monograph (xi -f 180 
pages) by B. Segre (Italian, England) on The nonsingular cubic 
surface. Solomon’s seal is here displayed in all its “complicated 
and many-sided symmetry” — in Cayley’s phrase — as never 
before. The exhaustive enumeration of special configurations 
provides an unsurpassed training ground or ‘boot camp’ for 
any who may wish to strengthen their intuition in space of three 
dimensions. The principle of continuity, ably seconded by the 
method of degeneration, consistently applied, unifies the multi- 
tude of details inherent in the twenty-seven lines, giving the 
luxuriant confusion an elusive coherence which was lacking 
in earlier attempts to “bind the sweet influences” of the thirty- 
six possible double sixes (or ‘double sixers,’ as they were once 
called) into five types of possible real cubic surfaces, containing 
respectively 27, 15, 7, 3, 3 real lines. A double six is two sextuples 
of skew lines such that each line of one is skew to precisely one 
corresponding line of the other. A more modern touch appears 
in the topology of these five species. Except for one of the 
three-line surfaces, all are closed, connected manifolds, while 
the other three-line is two connected pieces, of which only one 
is ovoid, and the real lines of the surface are on this second 
piece. The decompositions of the nonovoid piece into generalized 
polyhedra by the real lines of the surface are painstakingly 
classified with respect to their number of faces and other char- 
acteristics suggested by the lines. The nonovoid piece of one 
three-line surface is homeomorphic to the real projective plane, 
as also is the other three-line surface. The topological interlude 
gives way to a more classical theme in space of three dimensions, 
which analyzes the group in the complex domain of the twenty- 
seven lines geometrically, either through the intricacies of the 
thirty-six double sixes, or through the forty triads of com- 
plementary Steiner sets. A Steiner set of nine lines is three sets 
of three such that each line of one set is incident with precisely 
two lines of each other set. The geometrical significance of 
permutability of operations in the group is rather more com- 
plicated than its algebraic equivalent. The group is of order 
51840. There is an involutorial transformation in the group for 
each double six; the transformation permutes corresponding 



lines of the complementary' sets of six of the double six, and 
leaves each of the remaining fifteen lines invariant. If the double 
sixes corresponding to two such transformations have four 
common lines, the transformations are permutable. If the 
transformations are not permutable, the corresponding double 
sixes have six common lines, and the remaining twelve lines 
form a third double six. Although the geometry'- of the situation 
may' be perspicuous to those gifted with visual imagination, 
others find the underlying algebraic identities, among even so 
impressive a number of group operations as 51840, somewhat 
easier to see through. But this difference is merely one of ac- 
quired taste or natural capacity', and there is no arguing about 
it. However, it may' be remembered that some of this scintillating 
pure geometry was subsequent, not antecedent, to many' a 
dreary page of laborious algebra. The group of the twenty'- 
seven lines alone has a somewhat forbidding literature in the 
tradition of the late nineteenth and early twentieth centuries 
which but few longer read, much less appreciate. So long as 
geometry' — of a rather antiquated kind, it may be — can clothe 
the outcome of intricate calculations in visualizable form, the 
Solomon’s seal of the nineteenth century will attract its de- 
votees, and so with other famous classics of the geometric 
imagination. But in the meantime, the continually advancing 
front of creative geometry will have moved on to unexplored 
territory of fresher and perhaps wider interest. The world some- 
times has sufficient reason to be weary' of the past in mathe- 
matics as in everything else. 

What is geometry? 

In the typical episode of the nineteenth century' just recalled, 
we see once more the continual progression from the special to 
the general, in the emergence of widely applicable methods from 
laboriously' acquired collections of individual theorems, that 
characterized mathematics since 1800. The methods generalized 
from Cremona’s have retained their vitality' and interest, 
although the particular theorems in which they originated may' 
have lost their attractiveness for a generation trained in newer 
habits of thought for which those very theorems were partly 
responsible. So again, in seeking the things that have endured in 
mathematics, we are led to processes and ways of thinking rather 
than to their products in any one epoch. Wc shall see also that 
the conception of geometry' itself changed with time, until 


what was called geometry in one stage of the development would 
hardly have been recognized as such at an earlier stage. 

Attempts to obtain from professional geometers a statement 
of what geometry is are likely to be only nebulously successful. 
Beyond agreeing, more or less, on “geometry is the product of 
a particular way of thinking,” few geometers will commit 
themselves to anything less hazy. Accepting this for the present 
— we shall return to it in other connections — we assume that 
it has a meaning which can be ‘felt,’ if not understood; and we 
shall describe some of the main contributions of geometric 
thought to mathematics as a whole. 

Numerous representative selections might be made; the 
topics described here were chosen as an irreducible minimum 
on the advice of men actively engaged (1945) in developing the 
geometry of the twentieth century. The principal topics are: 
the vindication of Euclid’s methodology in the creation of 
non-Euclidean geometries, and the outgrowths of this in the 
modern abstract or postulational method; differential geometry 
from Euler, Monge, and Gauss to Riemann and his successors, 
with its profound influence on the cosmology and mathematical 
physics of the twentieth century; the principle of duality in 
projective geometry, and its final elucidation in the inventions 
(1831) of J. Plucker (1801-1868, German), also this most original 
geometer’s conception of the dimensionality of a space; Cayley’s 
reduction (1859) of metric geometry to projective; algebraic 
geometry, particularly its connection with Cremona (1863) and 
birational transformations and the analysis of abelian functions; 
Klein’s program (1872) for the unification of the diverse geome- 
tries existing in his day, and the supersession of this program 
after 1916; and finally, the abstract spaces and topology of the 
twentieth century, w'-hich some believe to be the beginning of a 
new type of mathematical thinking. 

Naturally, only a bare hint of so vast a territory can be given 
in the space at our disposal; and here as elsewhere in this ac- 
count we shall note only general trends. A history of any one 
of the topics would fill a book larger than this. The little de- 
scribed, however, may stimulate some to find out more about 
the subjects mentioned. Three of the topics are best considered 
under analysis, where they will be noted. Klein’s program, its 
successors, and topology are described in the chapter on invari- 
ance; what little can be said about the connection between 
algebraic geometry and analysis is deferred to the theory o 



functions of a complex variable; and the rise of theories of 
abstract space is followed in a later chapter as a consequence 
of the trend toward general analysis, first plainly noticeable in 
1906, for which mathematical physics was partly responsible. If, 
after all this, we are still unable to state what geometry is, 
we shall at least have caught a glimpse of the mathematics 
created by geometers in the worship of their inexplicit ideal. 

Euclid cleared of all blemish 

In 1733 the Jesuit logician and mathematician G. Sacchcri 
(1667-1733) completed his involuntary masterpiece, Euclidcs ab 
omni naevo vindicalus, in which he undertook to prove that 
Euclid’s system of geometry, with its postulate of parallels, 
is the only one possible in logic and experience. His brilliant 
failure is one of the most remarkable instances in the history of 
mathematical thought of the mental inertia induced by an 
education in obedience and orthodoxy, confirmed in mature 
life by an excessive reverence for the perishable works of the 
immortal dead. With two new geometries, each as valid as 
Euclid’s, in his hand, Sacchcri threw both away because he was 
willfully determined to continue in the obstinate worship of his 
idol despite the insistent promptings of his own sane reason. 

To ‘prove’ Euclid’s parallel postulate, Sacchcri constructed a 
bircctnngular quadrilateral by drawing two equal perpendiculars 
AD, BC at the ends A, B of a straight-line segment AB, and on 
the same side of AB. Joining D, C, he proved easily that the 
angles ADC, BCD are equal. The parallel postulate is equivalent 
to the hypothesis that each of ADC, BCD is a right angle. To 
‘prove’ the postulate, Sacchcri attempted to show the absurdity 
of each alternative. 

If each angle ADC, BCD is acute (‘hypothesis of the acute 
angle’), it can be proved that the sum of the angles of any 
triangle is less than two right angles; if each angle is obtuse 
(‘hypothesis of the obtuse angle’), the same sum is greater than 
two right angles; if each angle is a right angle, the same sum 
is equal to two right angles. Determined to establish the third 
possibility, Sacchcri deduced numerous theorems from each 
of the first two hypotheses, hoping to reach a contradiction in 
each instance. He disposed of the hypothesis of the obtuse 
angle by tacitly assuming that a straight line is necessarily of 
infinite length. The hypothesis of the acute angle was rejected 
by an improper use of infinitesimals. 


Cleansed by faulty reasoning of all blemish, Euclid’s geome- 
try shone forth to its worshiper as the absolute and eternal 
truth, the one possible mathematics of space. Saccheri died 
happy, unaware that he had proved several theorems in two new 
geometries, each as sound logically as Euclid’s. The devout 
geometer had unwittingly demonstrated that his unique idol 
was but one-third of a trinity, coequal with the others but not 
coeternal; for no geometry is the everlasting truth that Saccheri 
thought he had proved Euclid’s to be. It seems rather strange 
that the good geometer should have rejected the hypothesis 
of the obtuse angle so confidently; but here perhaps the fault 
was Euclid’s with his meaningless definition of a straight line. 
With a precise definition 3 of a straight-line segment as the 
shortest 4 distance between two points, the concept of a geodesic 
on a surface is almost immediate. The geodesics on a sphere 
(which is a surface of constant positive curvature) are arcs of 
great circles, the analogues of the ‘shortest distances’ in a 

plane. It is just possible, however, that Saccheri’s disciplined 
education required him to believe in a flat earth. 

Any geometry constructed on postulates differing in any 
respect from those of Euclidean geometry is called non-Eucli- 
dean. Saccheri’s two rejected specimens were the first non- 
Euclidean geometries in history. Ever since the time of Euclid, 
geometers had tried to deduce the parallel postulate from the 
others of Euclid’s system. No useful purpose is served by 
cataloguing scores of failures to achieve the impossible, although 
several disclosed interesting equivalents of the doubtful postu- 
late. A bibliography 5 of non-Euclidean geometry up to 1911 lists 
about 4,000 titles of books and papers by about 1,350 authors; 
and since 1911 the subject has expanded enormously. Much of 
the more recent work was directly inspired by physics, especially 
general relativity. Of the tentative steps toward a recognized, 
valid, non-Euclidean geometry between Saccheri’s (1733) and 

Lobachewsky’s (1826-9), we need recall only two. 

In 1766, J. H. Lambert (1728-1777, German) noted that the 
hypothesis of the obtuse angle is realized on a sphere, and 
remarked that a novel kind of surface would be required to 
represent the plane geometry corresponding to the hypothesis of 
the acute angle. Nothing came of this suggestion till 1868, 
when E. Beltrami (1835-1900, Italian) showed that the surface 
vaguely conjectured by Lambert is the so-called pseudo-sp iere. 
This is the surface of constant negative curvature generate 



by the revolution of a tractrix about its axis; it had been noted 
by Gauss, but without application to non-Euclidean geometry. 
But this belongs to the modern development, and we shall note 
its peculiar significance later. 

The first indisputable anticipation of non-Euclidean geome- 
try was by Gauss. As a boy of twelve, Gauss recognized that the 
parallel postulate presented a real and unsolved problem; but 
not till he was well past twenty did he begin to suspect that this 
postulate cannot be deduced from the others of Euclidean 
geometry. It is not definitely known when Gauss undertook the 
creation of a consistent geometry without Euclid’s fifth postu- 
late. It is certain, however, that he was in possession of the 
main results of hyperbolic geometry (Klein’s designation for the 
system constructed on the hypothesis of the acute angle) before 
N. I. Lobachewsky (1793-1856, Russian) published his complete 
system (1829), and therefore also before J. Bolyai (1802-1860, 
Hungarian) permitted his to be printed (1833) as an appendix of 
twenty-six pages in a semiphilosophical two-volume elementary 
mathematical work ( Tcntamen , etc.) by his father. 

It used to be claimed on the flimsiest circumstantial evidence 
that J. Bolyai had been influenced by Gauss. As it is now 
generally admitted that there are no grounds whatever for this 
hypothetical action at a very great distance, we shall pass it 
with the fact that J. Bolyai’s father, W. Bolyai (1775-1856), 
was a close friend of Gauss during his student days at the 

Gauss never made any public claim for himself as an inventor 
of non-Euclidean geometry. His anticipations of a part of what 
Lobachewsky and J. Bolyai accomplished, almost simultaneously 
and independently, were found in his papers after his death. 
Although he himself refrained from publishing the revolutionary 
geometry, Gauss encouraged others to proceed in their efforts 
to construct a consistent non-Euclidean system. Two of his 
correspondents made considerable progress: F. K. Schwcikart 
(17S0-1S59) and F. A. Taurinus (1794—1874), both German. 
The latter in particular obtained correct and unexpected results 
(1825-6) in non-Euclidean trigonometry. The earlier date 
coincides with that at which J. Bolyai is supposed to have con- 
vinced himself that hyperbolic geometry is consistent; the 
later with Lobachcwsky’s first paper, unaccountably lost by 
the Kazan Physico-Mathcmatical Society, on the new (hyper- 
bolic) geometry. For reasons that arc not exactly clear, Taurinus 


destroyed all copies of his own work on which he could lay his 

In the bald historical statement that Lobachewsky in 1826-9 
and J. Bolyai in 1833, almost simultaneously and entirely 
independently, published detailed developments of hyperbolic 
geometry, we have recalled one of the major revolutions in all 
thought. To exhibit another comparable to it in far-reaching 
significance, we have to go back to Copernicus; and even this 
comparison is inadequate in some respects. For non-Euclidean 
geometry and abstract algebra were to change the whole outlook 
on deductive reasoning, and not merely enlarge or modify 
particular divisions of science and mathematics. To the abstract 
algebra of the 1830’s and the bold creations of Lobachewsky 
and Bolyai can be traced directly the current (1945) estimate of 
mathematics as an arbitrary creation of mathematicians. 
In precisely the same way that a novelist invents characters, 
dialogues, and situations of which he is both author and master, 
the mathematician devises at will the postulates upon which he 
bases his mathematical systems. Both the novelist and the 
mathematician may be conditioned by their environments in 
the choice and treatment of their material; but neither is 
compelled by any extrahuman, eternal necessity to create 
certain characters or to invent certain systems. Or, if either is so 
conditioned, it has not been demonstrated that he is; and to an 
adult twentieth-century intelligence the multiplication of super- 
fluous and mystical hypotheses is a pursuit even more futile 
than It was in the days of Occam. 

In reporting this estimate of mathematics by informed 
opinion in 1945, we must also state that it was by no means 
universal. Many of the older generation still adhered to the 
Platonic doctrine of mathematical truths. Nor is there any 
reason to suppose that Plato will not again reign supreme in the 
minds of mathematicians. Less rational mysticisms than Plato’s 
have prevailed since the rediscovery of the virtues of blind 
irrationalism in 1914. But while the majority of mathematicians 
still believe they can see through an ancient fraud to the nonen- 
tity behind it, we shall record briefly how the humanization of 
mathematics came about. The deflation of older beliefs, how- 
ever, comprises the main contribution of non-Euclidean 
geometry to mathematical thought as a whole, and also, per- 
haps, the principal contribution of mathematics to the progress 
of civilization. For it seems improbable that our credulous 



race is likely ever to get very far away from brutehood until 
it has the sense and the courage to discard its baseless super- 
stitions, of which the absolute truth of mathematics was one. 

To appreciate fully the next item of more than local signifi- 
cance, we must describe a geometrical detail in each of four 
geometries; and we shall select that one, the existence of par- 
allels, which precipitated an apparently interminable deluge 
of non-Euclidean geometries after the hyperbolic geometry of 
Lobachewsky and Bolyai. In 1854, G. F. B. Riemann (1826- 
1866, German) invented a ‘spherical’ geometry, in which 
Sacchcri’s hypothesis of the obtuse angle is realized. The 
designations ‘hyperbolic’ and ‘elliptic’ refer to Cayley’s 
‘Absolute,’ to be noted later; Euclidean geometry is similarly 
called ‘parabolic.’ 

If P is any point in the plane determined by P and a straight 
line L not passing through P, there is, in parabolic geometry, 
precisely one straight line, L', through P which does not intersect 
L; L' is the unique parallel to L through P. In hyperbolic 
geometry, there are two distinct straight lines L r , L" through 
P, neither of which intersects L; moreover, no straight line 
through P and lying in the angle between V and L" meets L. 
Lobachewsky took L', L" as his parallels to L. In both parabolic 
and hyperbolic geometry two straight lines intersect in one 
point; Any two geodesics on a sphere (arcs of great circles) 
intersect in two points, and there arc no parallels. In Riemann’s 
so-called ‘spherical’ geometry, space is unbounded but finite; 
every ‘straight line’ (geodesic) is of finite length; and any two 
straight lines intersect twice, thus negating Euclid’s postulate 
that two straight lines cannot enclose a space. Riemann’s 
‘elliptic’ geometry can be visualized on a hemisphere if, as in 
his spherical geometry, ‘straight lines’ arc arcs of great circles 
of the whole sphere, and if the two extremities of an arc are 
regarded as the same point. Other ‘realizations’ are easily con- 
structed for all of the four geometries mentioned; our interest 
here is in the significance, or lack of it, of all such realizations. 
This marks the discontinuity in geometry as conceived before 
and after 1899. The three non-Euclidean geometries mentioned 
above arc usually called classical; ‘Ricmannian geometry,’ 
as used since 1916, is distinct from all these geometries. 

For about thirty years after the invention of hyperbolic 
geometry, few mathematicians paid any attention to it; and 
none, it appears now, foresaw what the non-Euclidean gcom- 



imagined space as a matrix of possible relations. Pasch all but 
eliminated both these conceptions in his restatement of geometry 
as a hypothctico-dcductive system in the tradition of Peirce. 
Instead of attempting to state definitions of points, lines, and 
planes, as Euclid had done, Pasch accepted these as the un- 
reduced elements of ‘space,’ and from postulated but unanalyzed 
relations between these atomistic concepts of his system pro- 
ceeded to deduction. 

The relations postulated were abstracted from the accepted 
geometric notions of centuries of working with diagrams. For 
example, it was explicitly stated as a postulate that two distinct 
points determine precisely one straight line. Pasch was therefore 
closer to Leibniz than to Newton; but he seems still to have 
believed in the existence of a ‘space’ in Newton’s sense. Whether 
or not he did, his was the first clear-cut presentation after 
Euclid’s of geometry as an exercise in postulational technique. 
Pasch went beyond Euclid in that he realized what he was doing, 
and did it deliberately, whereas Euclid seems to have been 
guided by visual imagery, and in consequence to have over- 
looked hidden assumptions. In any event, Pasch profoundly 
influenced the geometric thought of his contemporaries and 
successors. His conception of geometry, now an accepted com- 
monplace, met the usual opposition encountered by anything 
new and disturbing. Although neither the physical nor the 
philosophical meaning, if any, of ‘space’ was affected by the 
completely abstract reformulation of geometry, the novelty 
affected some — Veronese of Pascal’s hexagram for one — like a 
shocking blasphemy suddenly shouted in church. They quickly 
stopped their cars. Even if this new thing were consistent, it 
was too arid and too barren to be profitable mathematics. Geom- 
eters were permitted to retain their feeling for the ineffabilitics 
of the geometric mode of thought; but geometry itself was 
reduced to logical syntax. From a distance of fifty years it is 
difficult to sec why anyone got excited. Intuition and strict 
logical analysis can live in the same science without killing each 
other; and what one cannot do, the other can. 

The next man with courage enough to be unpopular was G. 
Pcano. At first he escaped notice except in Italy. But when 
(1 SSS) he began attempting to reduce all mathematics to a 
precise symbolism which left but few loopholes for vagueness, 
too-siippery intuition, and loose reasoning, he was regarded with 
•suspicion. With the help of several Italian collaborators, Pcano 



a small fraction of twentieth-century mathematics. But they 
were a potent catalyst for that mathematics, and they attracted 
hundreds of prolific workers. 

Among other subjects to profit by the revived and refined 
methodology of Euclid was non-Euclidcan geometry. One of the 
most curious geometries invented through a deliberate applica- 
tion of the postulational method was M. W. Dehn’s non- 
Archimcdcan system, which linked the similar triangles of the 
ancient Babylonians to the non-Euclidean geometry of the nine- 
teenth century. Suppressing the axiom of Archimedes, Dehn 
constructed a geometry in which similar triangles exist, and in 
which the angle-sum for any triangle is two right angles. But 
parallels are not unique as in Euclid’s geometry, an infinity 
being possible. Equally unforeseen consequences followed on 
applying the postulational method to projective geometry. 
The American geometers O. Vcblen and W. H. Bussey con- 
structed (1906) finite projective geometries in which a ‘plane’ 
contains only a finite number of ‘points’ and ‘lines.’ These 
finite geometries reduced the debates of the early nineteenth 
century on the ‘space’ of projective geometry to sequences 
of empty noises. Vcblen and J. W. Young produced (1907) a 
set of completely independent postulates (in the sense of E. H. 
Moore (1862-1932)) for projective geometry w'hich long remained 
a standard, and which must have convinced the most obstinate 
that geometry is a formal hypothetico-deductive exercise in 
logic. Americans probably did more to exploit the postulational 
technique after Hilbert’s Grundlagcn than their European 
colleagues, and their analysis was, on the whole, much sharper 
and clearer. The method was applied to algebra, geometry, 
arithmetic, topology, and other subjects by dozens of skilled 
mathematical logicians, among whom E. V. Huntington (1874—) 
may be specially mentioned for his exhaustive analyses of 
postulate systems in many fields. 

After the formal character of mathematics had been admitted 
by many, intellectual inertia proceeded once more on its time- 
honored course. If mathematics, and in particular geometry, is 
an arbitrary' creation of human beings, then surely the like is 
not so for traditional logic? In some extrahuman sense the 
logic which had lasted all of twenty-three centuries must be an 
absolute that not even mathematicians can defy. We shall 
sec in the proper place that this absolute, too, was abolished, 
but not till 1920. 


Returning to Saccheri (1733), we see now that after all he 
accomplished his purpose. In a sense that was to prove of in- 
comparably greater significance for the future of mathematics 
than a proof of the parallel postulate could ever have been 
Saccheri had cleared Euclid of all blemish. Although his work 
was ignored and forgotten for over a century after his death 
Saccheri deserves as much credit as Lobachewsky and Bolyai 
for having taken the decisive step toward the abolition of 
mathematical absolutism. 

The essential part of Euclid’s doctrine, 'strict deduction from 
explicitly stated assumptions recognized as such, began to come 
into its own only with the unconscious creation by Saccheri of 
non-Euclidean geometry. After more than two thousand years 
of partial understanding, the creative power of Euclid’s method- 
ology was gradually appreciated; and Saccheri, had he lived till 
1899, would have seen a profounder meaning than he intended 
in the title of his masterpiece, Euclides ab omni naevo vindicates. 
The import of what evolved from non-Euclidean geometry 
transcends geometry in epistemological value. The extramathe- 
matical by-product seems to stand a better chance of enduring 
than the vast accumulations of technical theorems which delight 
geometers, and which tomorrow may join the twenty-seven 
lines on a general cubic surface in the museum of mathematical 

A meaningless controversy 

A skilled geometer might devote his entire working life to 
one species of curves, say hypocycloids or bicircular quartics, 
and find something new and interesting to himself every day. 
But he would scarcely come upon general principles. If history 
is any criterion, generalizations in geometry have not been 
reached by piling theorem on theorem, but by purposeful efforts 
to slash through jungles of special results, or by equally con- 
scious attempts to find out why certain techniques furnish 
geometric theorems easily, while others demand more labor than 
their output justifies. A case in point is the prolonged struggle 
of the synthetic geometers during the first half of the nineteenth 
century to ascertain ‘why’ analytic methods were, apparently, 
so much more powerful than those of pure geometry. We sna 
trace the main lines of this fluctuating struggle in the following 
section. For the moment we consider the outcome, as this P°^ S1 ‘ 
bly is the item of greatest interest for mathematics as a w ho c. 



To state the conclusion first, analytic methods are more 
powerful than those of pure geometry because the experience of 
more than a century has shown them to be so. No philosophy of 
‘space’ and its ‘geometry’ has added anything of significance to 
this statement of brute fact; although the profane might suggest 
that as imaginary points, lines, etc., entered geometry through 
the formalisms of elementary algebra, and not through visual 
perceptions of diagrams, any attempt to disguise all algebraic 
concepts in an elaborate diagrammatic terminology could result 
only in unprofitable artificialities. This, at any rate, was the 
prevailing opinion among geometers in the 1930’s. They still 
continued to exercise their esoteric geometric intuition with 
telling effect, but only a few persisted in trying to modulate 
every uncouth configuration through delicately cadcnced in- 
volutions that would have ravished the pure geometric ear of 

It is conceded by all that the technical vocabulary of imagi- 
nary points, etc., is of great utility, and serves much the same 
purpose as the terminology of analysis. The statement that /(.v) 
is continuous at ,r = a , for example, condenses several assertions 
into one that can be used as a unit in deduction. It is unneces- 
sary to recur in each instance to the meaning of continuity, for 
its simpler implications have been worked out once for all, and 
can be applied without further thought. Similarly, the statement 
that a certain curve is a bicircular quartic, say, implies that each 
of the circular points at infinity is a node on the curve; and the 
standard elementary properties of nodes and the so-called 
circular points are applicable as units in investigating the curve. 

But this admitted utility of a geometrized algebraic vocabu- 
lary is not what distinguishes the geometers of the nineteenth 
century from the majority of their successors. The leading syn- 
thetic geometers of the past attempted to find the circular points, 
etc., in the ‘space’ of common experience, because they confused 
‘physical’ or ‘real’ or ‘a priori’ space — without attempting to 
explain what they understood by such space — with the abstrac- 
tions of their algebra and the inadequacies of their diagrams. 
The conception of geometry as a hypothctico-dcductivc system 
was about sixty years in the future when (1822) J. V. Poncclet 
(17SS-1S67, French) published his Trait c dcs prop rictes projec- 
tors ckr figures. This classic of the synthetic method was largely 
responsible for one of the most fruitful and, as is now widely 
believed, least meaningful contro%-ersics in the history of mathc- 


3 39 

SaratofF on the Volga, thinking out the pure geometry of his 
Traitr des proprietes projectiles des figures, which he published 
in 1822 after his return to France. He tells all, or nearly all, 
about it in his autobiographical introduction. The reservation, 
if justified, is hardly an argument in support of Poncclet’s main 
thesis that analytic methods are inferior to synthetic. For 
G. Darboux (1842-1917), another great French geometer, let 
the analytic cat out of the synthetic bag before a large and 
appreciative audience at the St. Louis (U.S.A.) congress in 
1904. According to Darboux, “We know, moreover, by the 
unfortunate publication of the SaratofF notes, that it tvas by 
the aid of Cartesian analysis that the principles which serve as 
the base of the Traite des proprietes projectives were first estab- 
lished.” As this blunt statement of fact seems not to be generally 
known to geometers, it may be that others besides Darboux liat'e 
considered it unfortunate. The proponents of analytic methods 
might consider this awkward disclosure extremely fortunate. 

But, by whatever means Poncclet first established his general 
principles, he put new life into a branch of geometry that was 
almost dead. Observing that certain properties of a plane con- 
figuration, such as the collincarity of three points in Pascal’s 
theorem, are unaltered by projection, Poncclet undertook a 
systematic investigation of such phenomena and defined the 
'graphic’ (our ‘projective’) properties of figures to be those 
which arc independent of the magnitudes (measures) of dis- 
tances and angles. As we shall sec later, Cayley forced the metric 
properties banned by Poncclet into a more inclusive projective 
geometry, in which the imaginarics that plagued Poncclet arc 
given a place of honor. Poncclet himself (1822), in accordance 
with his restatement of Kepler’s principle of continuity, intro- 
duced the line at infinity, and demanded in his plane geometry 
that every circle cut this line in the same two imaginary' points. 
All pairs of simultaneous equations representing circles were 
thus provided with the correct number of common solutions. 
G. Mange (1746-1S1S, French) had already used pairs of 
imaginarics to symbolize real spacial relations; but Poncclet 
was more concerned with attempting to find a ‘real’ justification 
for imaginarics in geometry. 

Poncclet’s principle of continuity amounts to the theorem of 
analysis, that if an analytic identity' in any' finite number of vari- 
ables holds for ail real values of the variables, it holds also by' 
analytic continuation for all complex values. The attempt to 


‘realize,’ or to disguise, this elementary fact of analysis in a sup- 
posedly existent space involved Poncelet in a furious controversy 
with Cauchy. The analyst insisted that the geometer’s reasoning, 
if not entirely illegitimate, was a needlessly complicated restate- 
ment of simple algebra; the geometer justified his tactics by 
proving numerous theorems with much greater ease than seemed 
possible by analysis. The dispute ended in a draw. But, as will 
be noted in connection with topology, Poncelet’s intuition out- 
ran his logic, and his attempted justification of his famous 
principle rested on a void. Nevertheless, he continued to derive 
correct geometrical theorems with astonishing facility. 

The principle of continuity was generalized in 1874-9 by 
H. Schubert (1848-1911, German), who went far beyond 
Poncelet’s boldest in his ‘calculus of enumerative geometry.’ 
Schubert’s ‘principle of the conservation of number,’ as the 
generalization was called, likewise rested on nothing that could 
now be recognized as a foundation. It asserted the invariance 
of the number of solutions of any determinate algebraic problem 
in any given numbers of variables and parameters under varia- 
tion of the parameters, or under substitution of special values 
for them, in such a manner that none become infinite, due 
account being taken of multiple solutions and solutions at 
infinity. This somewhat dangerous method was used with 
brilliant effect by several of the leading geometers of the nine- 
teenth century, including M. Chasles (1793-1880, French), J. 
Steiner (1796-1863, Swiss), Cayley, and J. G. Zeuthen (1839- 
1920, Danish) the last of whom profited by the more glaring 
oversights of his predecessors. Opinion on these subtle questions 
was still divided in 1945, the geometers affirming that their 
reasoning was sufficiently rigorous, the algebraists dissenting. 

All of the men just mentioned are conspicuous, and some 
famous, in the history of mathematics: Steiner, called by his con- 
temporaries “the greatest [pure] geometer since Apollonius, 
who could not write till he was fourteen; Chasles, a creative 
geometer and the judicious historian of geometry, whose 
Aperpu, historique sur I’origine et le developpement des methodes en 
geometric (1837) is still a classic of mathematical historiography, 
Cayley, the all-round mathematician, whose development 
(1846—) of the theory of algebraic invariants provided geometers 
with a new outlook on algebraic curves and surfaces; and last, 
Zeuthen, remembered in geometry and as an historian o 

Leaving the discredited principle of continuity, we pass o 



that of duality which, with its generalizations, left as substantial 
a residue of new and useful methods in geometry, algebra, and 
analysis as any mathematical invention of the nineteenth 
century. In its classic form, the principle scents to have been 
first clearly stated, but not fully understood, in 1825-7, by 
J. D. Gergonnc (1771-1859, French). Gergonnc noted that if 
in certain theorems of plane geometry the words ‘point* and 
‘straight line’ be interchanged, with corresponding changes for 
collincarities of points and intersections of lines, etc., inde- 
pendently provable ‘dual’ propositions result. He inferred that 
the original in all cases implies the dual, which therefore need 
not be proved independently. By this ‘principle of duality,’ 
Brianchon obtained his theorem as the dual of Pascal’s. Ger- 
gonnc also noted the corresponding principle in space of three 
dimensions, point and plane being duals, and the straight line 
self-dual. At one stroke an already vast empire of geometry was 
doubled in extent; and it was a self-denying geometer indeed 
who refrained from the practice, which quickly became epidemic, 
of publishing lengthy parallel columns of dual theorems. 

Gergonnc, like Poncclct, was a military man. Both laid 
claim to the principle of duality. Poncclct insisted that the 
principle was a consequence of the method of poles and polars, 
which he had used so brilliantly in his own geometry of conics; 
Gergonnc contended that poles and polars were not the root 
of the principle. Gergonnc was right, but a conclusive, simple 
demonstration of this fact was not forthcoming til! Plucker 
attacked the question algebraically, and gave the usual explana- 
tion by means of point and line coordinates to be found in most 
textbooks on projective geometry. But although Gergonnc was 
right, Poncclct was not entirely wrong. His contention was all 
but saved by the fortunate circumstance that the order and 
class of a conic each equal 2. 

Failing to make Gergonnc withdraw his claim to the prin- 
ciple by fair means, Poncclct resorted to foul, and succeeded in 
demonstrating that although a geometer may have been an 
officer, he is not necessarily a gentleman. The campaign of 
personal abuse and defamation of character which Poncclct 
waged against the comparatively obscure Gergonnc makes 
the Ncwton-I.cibniz controversy look like a love feast in Arcady. 
Today the law would interfere; but in the heroic age of projective 
geometry, mathematicians were free to persecute their defence- 
less enemies like the heroes some of them had been. 

The next great contribution to all mathematics for which 


Poncelet’s methods were partly responsible is in Vieta’s tradition 
of transformation and reduction. If a reversible transformation 
between the respective systems of coordinates in two spaces or 
between different coordinate systems in the same space, can be 
established, theorems in either system can be translated im- 
mediately into theorems in the other, the correspondence pro- 
viding the bilingual dictionary. If the restriction that the 
transformation be reversible is removed, the dictionary reads 
one way only, say like French into English, and configurations 
in the first space are mapped onto others in the second, but not 
vice versa. In either case there may be certain singular loci 
which must be excluded from the statements of theorems; an 
example will be given in connection with birational transforma- 
tions. There is no limit other than practical utility to the trans- 
formations that might be constructed in accordance with these 
very general specifications. If easily manipulated transforma- 
tions which alter the order of a curve (or surface, etc.) can be 
produced, the geometric gain is obvious. 

Numerous useful transformations of the kinds described 
have been constructed since Poncelet, in his method of reciprocal 
polars, first made a geometric element of one kind (a point) 
correspond to one of another kind (a line). Possibly the most 
extensively investigated transformations have been the bira- 
tional, in which the coordinates for either of two spaces are 
rationally expressible in terms of those of the other. This 
allusion must suffice here, as birational transformations are 
best described in connection with the theory of algebraic 
functions and Riemann surfaces, and must be deferred to a 
later chapter. 

Continuing with our selection of great principles that 
originated in projective geometry, we note next one of the 
least expected and most surprisingly simple generalizations in 
the evolution of mathematics, Pliicker’s theory of spacial 
dimensionality. We have already seen that Cayley (1843) and 
Grassman (1844) independently arrived at the notion of n-dimen- 
sional space, and that the latter defined the 7 z-dimensional 
manifolds which were to play a capital part in Riemann s 
geometry. Pliicker in his Analytische-geometrische Entioickeluitgin 
(1831) observed not only the analytic meaning of duality but 
also the germ of an idea which was to generalize the duality 
of Gergonne and Poncelet far beyond the obvious resources 
of pure geometry. He noted that the general equation o a 



straight line in plane Cartesian geometry contains two variables 
and two parameters, and that both the variables and the 
parameters enter the equation linearly. If the roles of the 
variables and parameters arc interchanged, the equation 
becomes that of a point. It is to be noticed that the number of 
parameters is the same, two, in both cases: the plane is a two- 
dimensional space, or manifold of two dimensions in points and 
in lines; and we say that the plane contains points and «> 2 
lines. These simple observations were the origin of Pluckcr’s 
vast generalization. 

If a class of elements is such that a unique element of the 
class is specified when any particular numerical values arc 
assigned to each of n numerical-valued parameters, the class 
is called a number-manifold, or a number-space, of n dimensions, 
and is said to contain <»" elements. The n parameters written 
in a prescribed order are called the coordinates of the general 
element of the class. For example, the general equation of a 
conic in the plane of Cartesian coordinates contains precisely 
five parameters; a particular set of values of the parameters 
specifies a unique conic; the plane is therefore a five-dimensional 
manifold, or space, when conics arc taken as the basic space- 
elements, and it contains °o & such elements. If this seems 
strange at first, it is no more so than the commonplace that the 
plane is a two-dimensional manifold of points. In Pluckcr's 
geometry, dimensionality is not an absolute attribute of space, 
but depends upon the basic elements constituting the space. 
A Cartesian plane, for example, is three-dimensional in circles. 
All of this was generalized early in the twentieth century to 
spaces in which the values of the parameters are not necessarily 
numbers; the resulting geometries are those of the various 
abstract spaces intensively studied since Frcchct’s creation 
(1906) of the first. These will be noted in other connections. 

Generalizing the classic duality for configurations of points 
and lines in plane geometry, Piuckcr stated a principle of 
duality for any two classes of configurations that have equal 
dimensionalities and arc both linear in the respective coordinates, 
equal in number, determining the common dimensionality. 
Each of the classes is a number-space as already defined. The 
‘geometry’ of each may be interpreted in many ways. For 
example, in the three-dimensional space constituted by all 
circles in a Cartesian plane, an equation between the three 
coordinates of a circle defines a family of cc 3 circles. We might 


proceed, as many did, to investigate in detail the properties of 
such families, defined by equations of degrees 1, 2, ... * 
Very simple plane representations, easily visualized, of the 
classical non-Euclidean geometries have been constructed from 
families of circles. Precisely as in the familiar Cartesian geometry 
in which points are the basic elements, in Plucker’s geometry 
we translate the algebra of systems of equations into properties 
of families of straight lines, conics, cubics, and so on. The duality 
which Gergonne and others believed to be an absolute attribute 
of ‘space’ peculiar to the intuitive, diagramed ‘space’ of ele- 
mentary projective geometry appeared in Plucker’s geometry 
as a trivial consequence of an unnecessarily restricted way of 
choosing systems of coordinates. 

Plucker’s abandonment of the deceptions of visual intuition 
for the explicitness of algebra and analysis finished something 
the classical non-Euclidean geometries had only half done. The 
arbitrary freedom in the mathematical construction of ‘spaces’ 
and ‘geometries’ at last made it plain that Kant’s a priori space 
and his whole conception of the nature of mathematics are 
erroneous. Yet, as late as 1945, students of philosophy were still 
faithfully mastering Kant’s obsolete ideas under the delusion 
that they were gaining an insight into mathematics. As Kant 
appealed to his mathematical misconceptions in the elaboration 
of his system, it is just possible that some other parts of his 
philosophy are exactly as valid as his mathematics. Against 
this it has been contended that Kant’s mathematics remains 
‘true’ in a higher realm of ‘truth’ beyond the comprehension of 
professional mathematicians, whose grudging science leaves 
them but little time to explore the really important questions 
of their subject. The difference of opinion may well be left there. 
The most significant residue of Plucker’s work was the demon- 
stration once more that geometry as practiced by geometers 
is an abstract, formal discipline. It should not be necessary to 
repeat that any experiences which may have suggested a par- 
ticular set of assumptions for a geometry are irrelevant to the 
mathematical development. . 

On the strictly technical side, Plucker elaborated (1865) m 
great detail the geometry of what we ordinarily picture as a 
Cartesian three-dimensional space, the point-space of elementary 
solid geometry and rigid bodies, but with straight lines instea 
of points as the basic elements. Since the equations of a genera 
straight line in the Cartesian space involve exactly four m e 
pendent parameters, Plucker’s ‘line geometry’ is that of a our 



dimensions! 'pace. Equations between the coordinates represent 
various families of straight lines; a family of » : straight lines 
is called a range, one of « 2 , a congruence, and one of s’, a 
complex; and these species are further classified according to the 
decrees of algebraic equations in the four line coordinates. The 
theory then proceeds partly by analogy frith the familiar geome- 
try bared on point coordinates in point-spaces of two and three 
dimensions. For example, instead of the cuacric surfaces 
defined by the general equation of the second degree in point 
coordinates ( x. y, z), the geometric configurations defined by 
an equation of the second degree in line coordinates (p, q. r, s) 
are investigated and classified into types, analogously to the 
classification of quadrics into pairs of planes, cones, cylinders, 
ellipsoids, hvpcrboioids, etc. This particular detail is the 
geometry of the quadratic line complex; the problem of classifi- 
cation led to much interesting algebra of the type associated 
with the reduction of matrices to canonical form. In line geome- 
try, a curve is visualized as an envelope of straight lines, not as 
a locus of points. 

The inevitable question, ‘What was the good of it all:/ is 
unanswerable. But for those who insist upon a scientific or 
industrial application for all mathematics, it may be recalled 
that Plucker’ s line geometry found an immediate interpretation 
in the dynamics of rigid bodies. A four-dimensional geometry 
of the late 1860’s thus justified its creation and its existence to a 
generation that believed in machinery with all its heart, with all 
its mind, and with all the soul it had. 

Synthesis versus analysis 

Returning now to 1827, when A. F. Mobius (1790-1860, Ger- 
man) introduced homogeneous coordinates in his highly original 
work. Der IcryeerJrische Calcul, we shall follow the struggle of 
synthetic methods against coordinates until both emerged 
victorious 5n the 1860’s. Although no sharp line divided the 
contestants or separated the geometers of different nationalities, 
the prolific Italian school preferred synthesis after the 1860*s, 
while the majority of French, German, and British geometers 
made greater me of analysis. At the beginning of the period, 
Steiner was the unapproachable champion of synthesis: Plucker, 
the unrivaled master of analysis. 

Plucker is usually regarded as the true founder of the method 
of homogeneous coordinates, in his Ar.alyiiscke-geomeirisckf 
r.r.tur:cke lunger. 6IS2S, 18:1). which also exploited abridged 


notation, usually attributed to E. Bobillier (1797-1832, French) 
and the simple analytic equivalent of Gergonne’s principle of 
duality. This was followed in 1835 by the System der analytische 
Geomelrie, in which, incidentally, curves of the third order were 
completely classified. Attempts by Cramer and others in the 
eighteenth century to discipline the swarms of curves of the 
fourth order had failed. In his Theorie der algebraischen Curven 
(1839), Plucker had a better success. This, however, was 
totally eclipsed by a discovery of a new kind and of the very first 
magnitude, ‘ Plucker’ s equations’ connecting the order, the class 
and the numbers of double points, double tangents, and points 
of inflection of algebraic plane curves. Cayley pronounced this 
discovery one of the greatest in the history of geometry. It be- 
came one of his life-interests to extend Pliicker’s equations to the 
singularities of skew curves (twisted curves, curves in space) 
and surfaces. G. Salmon, a fine geometer and algebraist who 
abandoned mathematics for theology, also made notable con- 
tributions to this elusive subject. It has extensive ramifications in 
modern algebra and analysis, the last through the theory of 
algebraic functions and their integrals. 

It was in this prolific third decade of the nineteenth century 
that line coordinates were invented. It is generally agreed that 
Plucker and Cayley imagined these coordinates independently. 
Many special surfaces of some interest were exhaustively investi- 
gated by both point and line coordinates. Two may be recalled 
as representative of the more interesting specimens collected 
and anatomized in this detailed sort of work: Kummer’s (1864) 
quartic surface, which is the so-called singular surface of the 
quadratic line complex, and which is represented parametrically, 
as Cayley was the first to discover (1877), by hyperelliptic 
theta functions; and the wave surface in optics, parametrized 
by elliptic functions. Through its generalizations to higher space, 
Kummer’s surface by itself generated an extensive department 
of geometry. Numerous French, German, Italian, and British 
geometers elaborated — perhaps overelaborated — this intncate 
specialty well into the twentieth century. But the general 
quartic surface in 1945 still presented unsolved problems, and 
possibly was too complicated for profitable attack by the weap- 
ons available. Interest in such matters had declined rapidly m 
the twentieth century, especially after 1920, and they seemed 
to belong definitely to a glorious but buried past. 

While Plucker, Cayley, and many others were creating 
modern analytic geometry with astonishing rapidity, the most 



ingenious pure geometer since Apollonius was engaged, with 
spectacular but severely restricted success, in attempting to 
forge synthetic geometry into an implement of what he hoped 
would be universal applicability. Steiner’s Systematised Ent- 
uickelung der Abhangigkeil geomctrischer Gesialtcn von cinander 
(1832) unified the classical methods of pure projective geometry 
and applied them with amazing skill to numerous special 
problems. Incidentally, Steiner proposed several theorems which, 
presumably, he had discovered by pure geometry, as challenges 
to be proved by other geometers. The last detail in one of these 
defied proof by analytic methods till the early 1900’s. 

The powerful method of inversion is also attributed to 
Steiner (1824), although others also invented it. It was inde- 
pendently come upon (1845) through physical considerations 
by W. Thomson (Lord Kelvin). He and other physicists applied 
it effectively in its synthetic form to electrostatics, calling it 
the method of images. Conversely, problems in potential theory 
can be disguised as exercises in inversion. The trick works both 
ways because inversion is a conformal transformation. 

Inversion was one of the first non-linear transformations to 
be studied deeply in geometry, although quadratic and cubic 
transformations had been familiar in algebra since the sixteenth 
century. Special birational quadratic transformations were 
used by Poncelet (1822), Plucker (1830), Steiner (1832), and 
systematically (1832) by L. I. Magnus (1790-1861, German). 
As a detail of historical interest, Magnus (1833) compounded 
two quadratic transformations to obtain a quartic transforma- 
tion, by which straight lines correspond to quartic curves. He 
was thus enabled to read off theorems on quartic curves from 
their images in straight lines. In somewhat the same direction, 
E. dc Jonquieres established (1859) a special correspondence 
between straight lines and curves of order n with a prescribed 
multiple point of order r. — 1 . Geometers are interested in 
these historical minutiae because Cremona, who in 1863 set 
himself the problem of determining all birational transforma- 
tions of order r. between the points of two planes, apparently 
overlooked this earlier work, and as late as 1861 believed that 
if n > 2 no such transformations exist. Geometers point out 
that had Cremona been conversant with the algebraic notion of 
closure, as in a group, he would have drawn the correct inference 
immediately from what was already well known. However, when 
he realized his oversight, he made rapid progress. What amounts 
to a capital theorem in the particular birational transformations 


named after him will appear in another guise when we discuss 
algebraic functions. It suffices to note here that M. Noether 
J. Rosanes (German) and W. K. Clifford (1845-1879, English) 
proved almost simultaneously (1870) that a Cremona trans- 
formation can be generated by compounding quadratic trans- 
formations. The prolific Italian school, from Cremona in the 
nineteenth century to Severi in the twentieth, developed the 
resulting algebraic geometry mainly by geometric methods; 
and indeed the corresponding algebra and analysis quickly 
become unmanageable. The permanent gain from all this 
somewhat confused development appears to be the methodology 
of establishing correspondences between classes of different 
types of geometric configurations. 

Another extensive division of geometry that developed from 
the geometry and analysis of the 1820’s-30’s is concerned with 
the intersections of a variable curve with the curves of a linear 
series; yet another, with the geometric properties of the inter- 
sections of two plane curves; and still another, with geometry 
on curves and on surfaces; and last, the representation of one 
curve or surface upon another. Parts of these advanced theories 
belong to algebraic geometry, parts to analysis, the latter 
through the parametric representation of curves and surfaces by 
means of certain special functions intensively studied during 
the nineteenth century. Nothing further can be said here about 
these highly technical developments; a little more will appear by 
implication in connection with analysis. But in taking leave of 
them, we record that literally hundreds of men from the 1860’s 
to the 1930’s devoted the best years of their working lives to 
these kinds of geometry. Where so many did work of high 
quality, it would be invidious to single out individuals. One, 
however, was outstanding, the fertile and industrious R. F. A. 
Clebsch (1833-1872, German)). 

The decisive battle in the' war between the purists and the 
analysts lasted twelve years', from 1847 to 1860. The first date 
marks the publication of K. C. G. von Staudt’s (1798-1867, 
German) Geometrie der Lage; the second, the revised version of 
this ‘geometry of position’ in the same author’s devastating 
masterpiece Beitrage zur Geometrie der Lage (1856, I860).. 

It may be said at once that the uncompromising purist von 
Staudt drove the enemy frAm the field, but that the analytic 
geometers retired in good order with all their machinery intact. 
The victor was left to enjoy the fruits of his barren victory 
alone. In proving that geometry could, conceivably, get a on ~ 



without analysis, von Staudt simultaneously demonstrated the 
utter futility of such a parthenogenetic mode of propagation, 
should all geometers ever be singular enough to insist upon an 
exclusive indulgence in unnatural practices. This may not have 
been what von Staudt intended; it is merely what he accom- 
plished. If the total exclusion of algebra and analysis from 
geometry must result in any game as complicated and as 
artificial as von Staudt’s, then the game is not worth its candle, 
and geometric purity has cost more than a normal geometer 
should be willing to pay. None of this detracts from the merits 
of what von Staudt did. His purification of geometry remains 
one of the masterpieces of mathematical reasoning. Somebody, 
no doubt, had to do once for all what von Staudt did, whether 
it was worth doing or not. Its lasting contribution to mathe- 
matics is the unintended self-destruction of the ideal of total 
geometric purity. 

Observing that cross ratio involves the concept of distance 
in the line segments from which the ratio is compounded, and 
remarking that projective geometry professes to be concerned 
with those geometric properties that are independent of distance 
and angle, von Staudt proposed to cut the vicious circle by 
eliminating measures, and therefore numbers, from geometry. 
The root of the trouble seemed to be that coordinates or their 
numerical equivalents, presumably extraneous to projective 
geometry, were subtly implicit in all the classical developments 
of the subject. The program of von Staudt would reduce number 
to form, the exact opposite of what Pythagoras proposed and 
what Kronecker believed he had accomplished. If both von 
Staudt and Kronecker achieved their aims, number and form 
may be one. But it seems more plausible that whatever identity, 
if any, underlies both is merely an irreducible abstract structure 
of mathematical logic on which both are based. Such speculations 
as these, however, were far in the future when von Staudt 
purified geometry. His theory of what he called ‘throws’ gives 
a purclv projective algorithm for cross ratio and imaginaries. 
Most remarkably, the algorithm distinguishes between a com- 
plex number and its conjugate; conjugate imaginaries appear 
as the double points of an involution on a real straight line. 
It is interesting to note here a similarity between von Staudt’s 
mathematical thought and Dedekind’s: faced with a finite 
problem in arithmetic, Dedekind resorted to infinite classes 
in his solution: determined to expel imaginaries from geometry, 
von Staudt replaced them by infinities of real points. 


It is sometimes asserted that von Staudt was not wholly 
successful in his attempt to geometrize real and complex num- 
bers. The abstract geometries of the twentieth century would 
seem to support this contention. For although it may be possible 
to geometrize the numbers with which von Staudt was con- 
cerned, it seems unlikely that any algorithm whatever could 
reduce the elements of an abstract space to anything either 
more or less abstract than what they already are. The problem 
solved by von Staudt, if he did solve it, is of a kind that has been 
clearly formulated only by the modern postulational method 
which was not in existence in the 1850’s. 

Cayley encountered a problem of the same genus as von 
Staudt’s in his projective theory of metric geometry. This will 
be described shortly. Cayley’s projective equivalent (1859) 
of metric distance is based on cross ratio, and therefore involves 
the very notion of distance which it was designed to eliminate. 
Cayley himself was aware of this, but he did not attempt to 
remove the vicious circle. It is probably correct to say that 
neither the nature of von Staudt’s and Cayley’s problems, nor 
the logical analysis necessary for satisfactory solutions, was 
understood before the twentieth century. 

The struggle between the purists and the analysts, as typified 
in two of its heroes, illustrates certain general phenomena in 
the development of mathematical thought of more than geomet- 
ric interest. Pliicker’s career might form the basis for a study of 
mental inertia. Steiner’s contemporaries, as already noted, 
called him “the greatest geometer since Apollonius.” Some 
even substituted Euclid for Apollonius in their meed of admira- 
tion for Steiner’s synthetic genius. Pliicker was not called 
anything much; he was rather ostentatiously ignored by nearly 
all the elite of geometry. Or at least he personally felt that 
his fellow geometers were smugly indifferent to his work; 
and he abandoned mathematics for physics, where he is still 
remembered. Toward the close of his life, Pliicker emerged into 
the light again to compose his great treatise on line geometry, 
Neue Geometrie des Raumes gegrundet auf die Betrachtung der 
geraden Linie als Raumelemente , published posthumously (1868 
-9) under the sympathetic editorship of Klein. 

Pliicker’s return to geometry was partly occasioned by the 
warm appreciation of Cayley for his work. Cayley appears to 
have been the one first-rate mathematician who had an adequate 
conception of what Pliicker was doing for geometry. Steiners 
dazz ing brilliance blinded the majority to Pliicker’s incom- 



parably more massive achievements. Pliicker’s geometry was 
neither pretty nor — vile but just word — elegant as Steiner’s 
was. Steiner flaunted his incapacity for analysis, although some 
of his colleagues insinuated that “the old fox” knew a great 
deal more than he would admit and, like Poncelet in his funda- 
mental work, occasionally concealed in synthesis what he had 
discovered by analysis. But even if this is no more than a 
malicious canard, Steiner was a contemporary of Apollonius 
in this thinking. Apollonius would have understood Steiner’s 
geometry immediately and, with a few days’ practice, might 
even have beaten his modern rival at the ancient game. But to 
understand and appreciate what Pliicker was doing, Apollonius 
would have needed a new brain of a kind they did not produce 
in ancient Greece. 

If anyone in the nineteenth century is to be dubbed the 
greatest geometer since Apollonius, Steiner now seems to be an 
unlikely candidate for the honor. Yet fashion turned its broad 
back on Pliicker and favored Steiner with its sweetest, silliest 
smile. As has happened more than once in the history of mathe- 
matics, the man with new and fruitful ideas had to die before 
he might enjoy whatever satisfaction there may be in the esteem 
of one’s fellow workers. 

Projective metrics 

As the last of the major contributions of projective geometry 
to mathematical thought which we shall describe, we select 
Cayley’s reduction (1S59) of metric geometry to projective. 
Cayley gave details only for plane geometry; but with suitable 
modifications his method can be extended to space of any 
finite number of dimensions in which a numerical ‘distance 
function’ is defined for any pair of elements in the space. 

Abstracting the familiar intuitive properties of the distance 
between any two identical or distinct points in a plane, geometers 
lay down the following postulates for the distance, D(p , q), 
between the elements p, q of any space whose elements are 
p, q, r, . . . . (1) To any two elements p , q (identical or dis- 
tinct), there corresponds a unique real number, their distance, 
D(p, <?)• (2) D{p, p) ~ 0. (3) D{p, q) 0, if p, q arc distinct. (4) 
D(_p, q) - D(q, />). (5) D(p, q) + D{q, r) ^ D(p, r). The last 
is called the triangle (or triangular) inequality; it has already 
been noted in another connection, and will occur again. 

It was observed in effect by Cayley and E. Laguerre (183-4- 


1886, French) independently that these live postulates for 
distance have a solution D(p, q) in plane geometry other than 
the usual one giving the distance between two points as a func- 
tion of their coordinates by means of the Pythagorean theorem 
With the new definition of distance, and a corresponding one 
for angle, Cayley converted metric geometry with its usual 
definitions of distance and angle into a species of projective 
geometry. In short, he showed that the metric properties of 
Euclidean space can be reinterpreted as projective properties. 
Although the details are too technical for brief description, a 
hint may be given of Cayley’s approach. The quotations are 
from his sixth memoir on quantics (1859) and his own notes 
on it in his collected mathematical papers. 

. . . the theory in effect is, that the metrical properties of a figure are not 
the properties of the figure considered per se apart from anything else, but its 
properties when considered in connection with another figure, viz., the conic 
called the absolute.” “Metrical geometry is thus a part of descriptive [projec- 
tive] geometry, and descriptive geometry is all geometry, and reciprocally. . . . 

Regarding Cayley’s ‘all’ we must remember that he was 
writing in 1859. Cayley at first honored his ‘absolute’ with 
a capital ‘A,’ a deserved tribute to the magnitude of his inven- 
tion. But on learning that the Absolute was commonly used 
by metaphysical theologians to designate a certain extraspacial, 
extratemporal Entity, Cayley, who was a devout Christian, 
hastily descended to lower-case ‘a.’ Cayley’s absolute can be 

It may have been the additive property of collinear dis- 
tances that suggested Cayley’s projective distance and his 
absolute. For if p, q, r are collinear points, and if the straight- 
line segments pq , qr, pr are taken with their proper signs accord- 
ing to the usual rule, then pq -f- qr — pr. This resembles the 
theorem for the logarithm of a product. In any event, Cayley 
defined the distance D(p, q) between two points p, q in terms of 
a logarithm, as follows. The join of p , q cuts a certain fixed 
conic, Cayley’s ‘absolute,’ in two points p', q'; when p, ?are 
any fixed points, the four collinear points p , q, p q', taken m a 
certain order, determine a unique cross ratio; a constant, k, times 
the logarithm of this cross ratio is Cayley’s definition of D(p, ?)• 
It is easily seen that this D(p, q ) satisfies the stated postulates 
for a distance function. 

Thirteen years after Cayley’s reduction of metric properties 
to projective by means of his absolute, Klein (1871) notice 
that the projective definitions of distance and angle provided a 



simple unification of Euclidean geometry and the classical 
non-Euclidean geometries. These geometries, Klein showed, 
differ basically only in their respective distance functions. 
In Cayley’s definition, the constant k and the conic fixed as 
the absolute can be so chosen that the respective classical 
geometries of Lobachewsky and Bolyai, Riemann, and Euclid 
arc completely specified according as the absolute is real, 
imaginary, or degenerate. 

This striking result of Klein’s was a fitting climax to half a 
century’s striving for clarity in the projective geometry restored 
to life by Poncelet. Still greater things were to come a year 
later (1872) in Klein’s famous Erlangcr Programm. This will be 
noticed in connection with invariance. Klein’s program domi- 
nated much of geometry for almost half a century. It was super- 
seded by younger ideas that became popular only with general 
relativity after 1916, but which had their origin in Riemann’s 
revolutionary' work of 1854. We shall consider this next. 

From cartography to cosmology 

The problem of constructing fiat maps of the earth’s surface 
was one origin of differential geometry, which may be roughly 
described as the investigation of properties of curves and sur- 
faces in the neighborhood of a point. Still roughly, it is required 
to specify the geometry of a sufficiently small neighborhood with 
sufficient accuracy, the specification to be valid for the neighbor- 
hood of any point on the curve or surface investigated. Another 
origin of this ‘local’ geometry' was the study', in the seventeenth 
and eighteenth centuries, of tangents, normals, and curvature, 
the calculus having provided adequate means for a general 
attack. A third source is evident in the dynamics of the eight- 
eenth century, particularly in constrained motion, as in the 
dynamics of a particle restricted to move on a prescribed surface. 
With problems of these general types occur also their obvious 
inverses. For example: given a particular formula for the geo- 
desic distance between any' two neighboring points, to deter- 
mine the most general surface for which the formula holds; or 
to classify surfaces with respect to their lines of curvature. 
M any of these differential problems have immediate generaliza- 
tions to space of any finite number of dimensions. The resulting 
theories, as might be anticipated even from these meager 
hints, are of vast extent and have close connections with 
differential equations and mathematical physics. 

Attempting neither a history' nor a catalogue of what has 


been done since 1700 in differential geometry, we shall select 
a few typical incidents in the main line of progress, sufficient 
to connect the physical algebra already discussed with the 
analysis, differential equations, mechanics, mathematical physics 
and the non-Riemannian geometries of the twentieth century to 
be described in subsequent chapters. The increasing attention 
paid to quadratic differential forms from Gauss (1827) to 
Riemann (1854), Christoffel (1869), and Lipschitz (1870), then 
from Ricci (1887) to Einstein and others (1916—), blazes an 
easily followed trail from the cartography of the earth’s surface 
to the mapping of a large sector of cosmology on differential 
geometry. A map is not necessarily a picture on a sheet of paper. 
The maps of theoretical physics are mathematical descriptions 
of physical phenomena. 

Like so much else in modern mathematics, differential geom- 
etry got its first real start in the analysis of “the myriad-eyed 
Euler,” who overlooked nothing in the mathematics of his age, 
totally blind though he was for the last seventeen years of 
his life. In 1760 he investigated lines of curvature. This work 
inspired Monge to his own more systematic investigations (1781) 
in the same direction, and to his general theory of curvature, 
which he applied (1795) to the central quadrics. Equally signifi- 
cant for the future of mathematics was Monge’s elucidation 
of the solutions of partial differential equations by means of 
his theory of surfaces. The geometric language in which partial 
differential equations are frequently discussed originated in this 
early work of Monge. 

Another of Monge’s inventions, his descriptive geometry, 
is of less mathematical interest than his analysis of differential 

equations, but possibly of greater technological importance. 
Without descriptive geometry of some sort, the engineering 
sciences of the nineteenth century would have developed much 
more slowly than they did. Monge’s scheme for representing 
solids on one plane diagram by means of two projections, a ‘plan 
and an ‘elevation’ on two planes originally at right angles to 
each other before being laid fiat, facilitated the visualization 
of spacial relations, and provided a uniform graphics for solving 
such problems as determining the curves in which two or more 
surfaces intersect. Cut-and-try methods might waste a great 
deal of metal in fitting two pipes of different dimensions at a 
given angle. This problem is solved with no waste as one of t e 
earlier exercises in descriptive geometry. Practical mechanics 
drawing, without which the construction of modern machinery 



would hardly be feasible, evolved from Monge’s simple scheme. 
It is seldom pleasant to give the devil his due; but history com- 
pels us to state that a problem in fortifications was the origin 
(1763) of descriptive geometry. The French militarists thought 
so highly of Monge’s invention that they forbade him to pub- 
lish it, and for about thirty years kept it a secret for their 
own use. Monge’s account of the subject was first published in 
179 5~6. 

Continuing with what since about 1920 has been called 
classical differential geometry, we note the Applications de 
geometric ti de mechanique (1S22) of E. P. C. Dupin (1784— 1S73, 
French). Dupin’s work was prophetic in several respects. Al- 
though the indicatrix was not invented by Dupin, he made 
more effective use than had his predecessors of this suggestive 
conic in which a plane parallel to, and ‘infinitesimally near to,’ 
the tangent plane at any point of a surface intersects the sur- 
face. Analytically, the indicatrix introduces a quadratic differen- 
tial form into the geometry of certain curves (the asymptotic 
lines) on a surface. 

This is analogous to a method of approximation in mathe- 
matical physics, where the state of a medium in the neighborhood 
of a point is obtained to a sufficient degree of approximation 
by neglecting infinitesimals of order higher than the first in 
the Taylor expansion of the function expressing the exact state 
of the medium at any point. This procedure is not universal; 
but where it is applicable, it is one source of linear differential 
equations in the physical sciences. Geometrically, the indicatrix 
is useful in the study of two of the most interesting families of 
curves on surfaces, the asymptotic lines and the lines of curva- 
ture. Dupin also investigated triply orthogonal families of 
surfaces, not as a barren exercise in the differential calculus, but 
because certain instances of such families are of the first impor- 
tance in potential theory and other departments of mathematical 
physics. This aspect of differential geometry will be noted in 
another connection, when we follow the contributions of physics 
to mathematics, especially in Lame’s conception of coordinates. 
Another detail of Dupin’s geometry’ was to assume an unfore- 
seen significance in the lS90’s, when Klein and M. Bocher 
(IS67-19IS, U.S.A.) observed that the surfaces called cyclides, 
invented by Dupin, afford a unified geometric background for a 
wide class of differential equations of scientific importance. A 
cyclidc is the envelope of a family of spheres tangent to three 
fixed spheres. Dupin’s geometry was thus one source of much 


analysis of the nineteenth century. Triply orthogonal systems of 
surfaces, for instance, were the occasion for one of Darboux’ 
more famous works, extending to 56 7 pages, which in turn 
partly inspired G. M. Green (1891-1919, U.S.A.) to a notable 
simplification (1913) of the general theory as an application of 
the so-called projective differential geometry of E. J. Wilczynski 
(1876-1932, U.S.A.). The last is based in part upon a pair of 
simultaneous partial differential equations of the second order. 
Green’s 27 pages (1913) incidentally included the meat of 
Darboux’ 567. 

Projective differential geometry as practiced in the third 
decade of the twentieth century offered an interesting example 
of national preferences in mathematical technique. The two 
principal schools, the American and the Italian, sought essen- 
tially the same objectives, but by radically different methods. 
Each progressed far in its own direction; both were effectively 
halted, at least temporarily, by obstacles apparently' inseparable 
from their respective methods. Theoretically adequate for any 
problem that might arise in the subject, the American method 
was retarded by wildernesses of unavoidable calculations. A 
less prosaic but equally' discouraging difficulty, to be described 
presently', blocked the Italian approach to the generality of 
a projective differential geometry' of higher space. 

The American school followed the lead of Wilczynski, who 
presented his theory', with numerous applications to special 
problems, in a scries of memoirs, beginning in 1901, and in a 
treatise (1906) on the general method. Wilczynski had been a 
pupil of L. Fuchs (1833-1902, German), under whom he ac- 
quired a mastery of the theory' of differential equations as it 
was at the close of the nineteenth century'. It was therefore 
but natural that lie should base his geometry on a complete 
independent system of invariants and covariants of a sy'stemof 
one or more linear homogeneous differential equations. A funda- 
mental set of solutions of the equations uniquely determines 
the several geometric objects investigated, up to a projective 
transformation. Under appropriate transformations, of the 
dependent and independent variables in the differential equa- 
tions and in the parametric equations of the accompanying 
geometric objects, the objects and the forms of the differentia 
equations are invariant, although the coefficients of . the equa- 
tions will usually' be changed. The covariants basic, for. t e 
geometry' are functions of the new coefficients, their derivatives, 
and the new dependent variables, which differ at most by a 



factor from the same functions of the original variables and 
coefficients; an invariant is a covariant not containing the 
dependent variables or their derivatives. The Lie theory of 
transformation groups (described here in the chapter on in- 
variance) is the implement of calculation for obtaining the 
covariants and invariants as necessary preliminaries to the 
geometry. Probably almost anyone who has ever seriously 
attempted to solve differential equations by the Lie theory 
will appreciate the labor inherent in any such heroic project 
as Wilczynski’s and agree with Galois that, whatever the nature 
of its unchallenged merits, the theory of groups does not afford 
a practicable method for solving equations. Galois of course 
was speaking of algebraic equations, but his opinion, in the 
judgment of experts in the Lie theory, carries over to differential 
equations. Beyond a not very advanced stage of complexity, 
the calculations become prohibitive to even the most persevering 
obstinacy. The Italian method circumvented the Lie theory. 

About 1913 the Italian school headed by G. G. Fubini 
(1879-19-13) approached projective differential geometry through 
differential forms, arriving at systems of differential equations 
of the type from which Wilczynski had started. By restricting 
the analysis to systems in which the coefficients are legitimately 
specialized, and thereby simplified, by permissible transforma- 
tions, the basic covariants arc reduced to fairly manageable 
shape. The method of calculation is the absolute differential 
calculus, or tensor analysis, of M. M. G. Ricci (1853-1925, 
Italian), which was noted earlier in connection with the general 
progress of recent mathematics toward structure. The Ricci 
calculus, however, originated in the algebra of quadratic differ- 
ential forms. It was therefore inapplicable to the higher differen- 
tial forms hinted at in passing by Ricmann in his dissertation 
(1854) on the hypotheses which underlie geometry. But these 
forms arc those appropriate for a projective differential geometry 
of higher space. The Italian method seemed definitely to be 
inextcnsiblc to a variety of in dimensions in a space of n > 4 
dimensions, for 1 < m < n — 1. Nor is there a covariant 
quadratic form for these cases. It is noted in another connection 
that the lack of an absolute calculus for differential forms in 
higher space may be supplied, if, for example, physical specula- 
tions should render a serious effort to develop such a calculus 
scientifically profitable. The Ricci calculus did not come into 
its own geometrically until it was publicized by the relativists, 
when the geometers adopted and further developed it. The 


projective differential geometries of the American and Italian 
schools do not seem to have attracted physicists. 

These somewhat miscellaneous details have been recalled to 
underline the estimate of classical differential geometry v,i| c V, 
was that of a majority of professionals in the 1920’s. Since 
its inception in the work of Euler and Monge, differentia! 
geometry had expanded somewhat lawlessly, until by 1903 it 
embraced a loosely coordinated collection of special problems 
and incomplete theories, thrown together with no detectable 
aim and without any clearly defined objective. Such, for exam- 
ple, was substantially the opinion of Hadamard. In contrast 
with this disorderly luxuriance, the differential geometry that 
became popular with the application of Riemannian geometry 
to physics and cosmology in general relativity was unified ad 
given definite aims by the absolute differential calculus, or 
tensor analysis, of Ricci and Levi-Civita. When at last a uni- 
formity in method was recognized, interest in classical differ- 
ential geometry all but collapsed. Numerous special result- 
obtained in the older tradition had long since passed into the 
general structure of infinitesimal geometry and analysis; but 
creative work in differential geometry took a new direction. 
Among other special developments which had seemed promising 
in the early 1900’s, but which had lost much of their appeal bj 
the 1920’s, the projective differential geometry of one American 
school joined the classics which arc respected but seldom culti- 
vated. The line of descent from the old to the new, as already 
indicated, was from Gauss, through Ricmann, to the tensor 

Gauss (1827) made the first systematic study of quadratic 
differential forms in his Disquisitioncs gen cralcs circa 
curvas , in which the main theme is the curvature of surfaces. Tie 
forms investigated are in two variables only. With its relevance 
for the deformation of surfaces and the applicability of one 
surface on another, Gauss’ theory is a direct descendant oi 
cartography. This aspect, however, was not that which sug- 
gested the far-reaching generalization of differential geometry 
by Ricmann. Geodesy also was one of Gauss’ major interests 
(1843, 1847) in applied mathematics; and it too is part!) * 
matter of quadratic differential forms, the line element on a 
spheroid being the square root of a quadratic differential form 
in two variables with variable coefficients. Taking the final step 
in this direction, Ricmann, in one of the most prolific contn u 
tions ever made to geometry, passed immediately to the gene:<- 



quadratic differential form in n variables, with variable coeffi- 
cients, in his vital classic on the foundations of geometry, Ijbcr 
die llypothesen zvelche der Geometric zu Grundc liegen , 1854. 

Ricmann’s taste for speculative philosophy has made parts of 
his great essay needlessly difficult for mere mathematicians. 
Fortunately for geometry, Ricmann’s mysterious description of a 
manifold can be ignored; for when he proceeds to mathematics, 
he actually uses nothing more abstruse than an n-dimensional 
number-manifold. It would be interesting to know whether 
Ricmann imagined himself the originator of this notion. But as 
he seldom mentions other mathematicians in any of his work, 
even where it is plain that he has profited by their ideas, it is 
impossible to say how much, if anything, he owed to others. The 
general manifolds which Riemann attempted to define, but which 
he did not use, might be interpreted as the abstract spaces of the 
twentieth century. 

The mathematics of Riemann’s geometry interweaves two 
fundamental themes: a generalization of the Pythagorean 
theorem to any space (number-manifold) of n dimensions; 
curvature in such spaces. If 

, .Vn), (-V 1 ~f* dx 1 , . • . , X n -f- dXji) 

arc the coordinates of neighboring points in the space, and if ds 
is the infinitesimal distance between these points, it is postulated 
that ds- — B Zg,-jdx;dxj, in which the double summation refers 
to i, j t= I, . . . , ii] the £,7 are functions of #j, . . . , x n ; and 
£,7 — gj;. In laying down this postulate, Ricmann recognized 
that it gives a sufficient, but not necessary, specification of an 
elementary distance which is to retain the cardinal properties 
of a distance function; and he explicitly stated other possibilities. 
These had not been exploited (at least in print) as late as 1945, 
although as early as 1924 H. P. Robertson (1903-, U.S.A.) had 
investigated the analogue, for these possibilities, of the tensor 
calculus appropriate for the Ricmannian geometry of general 
relativity. The metric geometry of a particular Ricmannian 
space is determined by the g i} - occurring in the ds- for the space. 
Ignoring special cases, Ricmann proceeded at once to his gen- 
eralized curvature, guided partly by analogy with the Gaussian 
theory for a two-dimensional space. He then made the remark- 
able conjecture that his new metrics would reduce questions 
concerning the material universe and the “binding forces” 
holding it together to others in pure geometry. 

Bolder even than Riemann, Clifford confessed his belief 


(1870) that matter is only a manifestation of curvature in a 
space-time manifold. This embryonic divination has been 
acclaimed as an anticipation of Einstein’s (1915-16) relativistic 
theory of the gravitational field. The actual theory, however 
bears but slight resemblance to Clifford’s rather detailed creed! 
As a rule, those mathematical prophets who never descend to 
particulars make the top scores. Almost anyone can hit the side 
of a barn at forty yards with a charge of buckshot. 

The next long stride after Riemann’s toward modern differ- 
ential geometry was the determination by Christoffel (1869) of 
necessary and sufficient conditions that a quadratic differential 
form of the kind in Riemann’s ds 2 be transformable into another 
by a general functional transformation on the variables. The 
same problem was also treated by Lipschitz (1870). Christoffel’s 
solution proved the more useful. In the course of his analysis, 
Christoffel invented the process named covariant differentiation 
by Ricci (1887), and used it to derive a sequence of tensors from 
a given one. Beltrami and others, especially of the Italian school 
of geometers, used what are essentially tensors; but it remained 
for Ricci to isolate and perfect the tensor calculus as an inde- 
pendent algorithm. 

The algebra of tensors as a generalization of vectors was 
mentioned in an earlier chapter. The further development of 
Riemannian geometry will be noted in connection with invari- 
ance. We may conclude this sketch with a summary indication of 
the mathematical reason for the scientific utility of tensors. 

A functional transformation on the variables of a tensor 
transforms the tensor into another whose components are linear 
homogeneous functions of the components of the original tensor. 
A tensor, like an ordinary vector, vanishes if and only if each 
of its components vanishes. A transformation of the kind stated 
is, geometrically, a general transformation of coordinates, when 
the variables are interpreted as coordinates in a space of the 
appropriate number ojf dimensions. It follows that if a tensor 
vanishes in one system of coordinates, it vanishes in all; the 
homogeneity is the decisive factor. This is equivalent to saying 
that if a system of equations is expressible as the vanishing of 
a tensor, then the systcpm will be invariant under all transforma- 
tions of the variables in the system. But this is precisely the 
condition imposed by one of the postulates of general relativity 
on a system of equations, if the system is to be an admissible 
mathematical formulation of an observable sequence of events in 
physics or cosmology. 


The Impulse from Science 

This chapter is introductory to the six following, in which 
wc shall describe certain typical developments in the evolution 
of analysis from the seventeenth century to the twentieth. 
Analysis, perhaps more clearly than algebra or geometry, ex- 
hibits the constant influence of science on the general develop- 
ment of mathematics. 

We saw that the calculus owed as much to kinematics as it 
did to geometry'. From the death of Newton (1727) to the 
twentieth century, science continued to stimulate mathematical 
inventiveness. Of subsequent additions to mathematics originat- 
ing at least partly in science, the most highly developed are the 
vast domain of differential equations, the analysis of many 
special functions arising in potential theory and elsewhere, 
potential theory itself, the calculus of variations, the theory' 
of functions of a complex variable, integral equations and 
functional analysis, statistical analysis, and differential geom- 
etry. By 1800, the calculus of variations and differential equa- 
tions had advanced sufficiently to be recognized as autonomous 
but interdependent departments of mathematics; the statistical 
method was still an embryonic possibility in the theory of prob- 
ability: while the theory* of functions of a complex variable had 
yet to wait a quarter of a century for systematic development by 
Cauchy, although some of the basic results were implicit in the 
applied mathematics of Lagrange and others in the eighteenth 

In following the growth of rigor from 1703 to 1900, we noted 
a constantly sharpening precision of mathematical logic, and 
saw that attempts to provide a self-consistent foundation for 



analysis led in the early twentieth century to a period of con 
fusion and a recognition of the necessity for ever more subtle 
reasoning. Leaving all doubts behind for the present, we nor 
enter an untroubled region where the end justifies the mean" 
The end is the increase of scientific knowledge, to which mathe- 
matics is but one of several means. As in previous accounts ve 
shall attend only to typical features illustrative of general 
trends. There are first two matters of possibly wider significance 
to be noticed: the influence of eighteenth-century mathematics 
on society; and the response of society, especially after the 
Napoleonic era, to mathematical research. 

Mathematics in the Age of Reason 

The most significant contribution of eighteenth-century 
mathematics to civilization was a rational outlook on the phys- 
ical universe, for which dynamical astronomy and analytic 
mechanics were mainly responsible. 

The eighteenth century has been called the Age of Reason, 
also an age of enlightenment, partly because the physical science 
of that century attained its freedom from theology. In the 
hundred years from the death of Newton in 1727 to that of 
Laplace in 1827, dogmatic authority suffered the most devastat- 
ing of all defeats at the hands of scientific inquiry: indifference. 
It simply ceased to matter, so far as science was concerned, 
whether the assertions of the dogmatists were true or whether 
they were false. At the beginning of the period, it was customary 
to seek a teleological explanation for the principles of mechanics 
to accord with the orthodox theology of the time; when Laplace 
died, all such irrelevancies had been quietly ignored for forty 
years. Mechanics had at last come of age. Absolute truth, as 
revealed by science, fled to pure mathematics, where, according 
to some, it still resides. 

The French Revolution, beginning in 1789, accompanied the 
change; and we might be tempted to ascribe the maturing of the 
exact sciences wholly to that very thoroughgoing upheaval. But 
the final liberation had occurred in the preceding year with the 
long-delayed publication of Lagrange’s analytic mechanics. 
Here, for the first time, a masterpiece of mathematics an 
science of the very first rank stood erect on its own mathematics 
and scientific feet without external support. No mysterious 
spirit of nature was invoked; the work undertook to describe, not 
to explain, the mechanical behavior of material systems. 


A free translation of a lew sentences will indicate two of the 
respects in which Lagrange’s mechanics differed radically from 
its predecessors in both science and mathematics. In his preface, 
Lagrange writes: 

I have set myself the problem of reducing this science [mechanics], and the 
art of rohnng the problems appertaining to it, to general formulas, whose 
rimplc development gives all the equations necessary for the solution of each 
problem. . • . No diagrams will be found in this work. The methods which I 
expound in it demand neither constructions nor geometrical or mechanical 
reasonings, but solely algebraic [analytic] operations subjected to a uniform 
and regular procedure. Those who like analysis will be pleased to see mechanics 
become a new branch of it, and will be obliged to me for having extended its 

From this it is clear that Lagrange fully realized the signifi- 
cance of what he had done. The following quotation, typifying 
the spirit of the entire work, indicates his grasp of the abstract 
nature of mathematical mechanics: “The second fundamental 
principle of statics is that of the composition of forces. It is 
founded on this supposition: ...” Thus the principles of 
mechanics arc founded on suppositions , that is, on postulates, 
and arc not eternal truths revealed to a groping mankind by the 
grudging generosity of some supernatural intelligence. It is 
mathematical and scientific rationalism like this that validates 
the claim of the eighteenth century to be called an age of reason. 

Such clarity of mind as Lagrange’s, however, was the rare 
exception among mathematicians and scientists in his day and 
for over a century after his death in 1S13. Lagrange’s most 
prominent contemporary in the exact sciences, the self-confident 
Laplace, convinced himself and two generations of eager philoso- 
phers that the Newtonian mechanics of the heavens was abso- 
lutely and eternally true; and on this basis he sought to establish 
the everlasting stability of the solar system. 

Almost aggressively hostile to the pretensions of the older 
absolutism, the would-be skeptic Laplace substituted one dog- 
matic creed for another. It was largely due to the successes of 
his own celestial mechanics and his widely appreciated popular 
exposition of the mathematical consequences of Newtonian 
gravitation that a crude mechanistic philosophy afflicted nearly 
all physical scientists and many philosophers of the nineteenth 

Eighteenth-century mechanics was also partly responsible for 
the speed with which machinery overwhelmed civilization in the 
early nineteenth century. Instead of remaining the private 


servant of the intelligentsia, the mechanistic philosophy incon- 
tinently shared its inestimable benefits with the proletariat 
Hundreds of thousands to whom Lagrange and Laplace might 
have lectured for years with no transfer of ideas were converted 
by the dumb, inerring accuracy of their monotonous machines. 

That a fully developed abstract theory appeared first in 
modern mathematics from the applied side is less remarkable 
than it may seem at first sight. Mechanics and mathematical 
physics generally had no such crushing burden of tradition to 
throw off as had geometry*. Mathematical mechanics was little 
more than a century old when Lagrange saw what it was. Only 
about sixty years before Lagrange published his mechanics, 
Saccheri’s willful faith in the sanctity of Euclidean geometry 
had compelled him to ignore the promptings of his own acute 
reason. Possibly if Archimedes rather than Galileo and Newton 
had formulated the basic ‘laws’ of dynamics, Lagrange might 
have hesitated to deflate the foundations of his system to “sup- 
positions.” But the postulates of mechanics had not had time 
to fossilize into eternal truths, and Lagrange was not tempted to 
outrage his reason — at least in mechanics. But in his effort to 
rigorize the calculus by basing it on Taylor’s expansion, he was as 
tradition-bound as Saccheri, and possibly for the same reason. 
The problems of continuity were as old as those of geometry, and 
the same almost superhuman intransigency rvas demanded to 
flout tradition and advance in a totally new direction. 

The eighteenth century may indeed have been the golden age 
of reason in philosophy and human affairs generally that it is 
said to have been. The exact sciences, as we have just noted, 
also submitted to reason in that hard-headed century. But in 
pure mathematics, there was a marked decline from the standard 
which the ancient Greeks set themselves. The best that reason 
could do when confronted with a problem in continuity was 
Lagrange’s curious attempt to rigorize the calculus. No classic 
Greek mathematician could have deluded himself so completely 
as did Lagrange, the greatest mathematician of his age and one 
of the greatest of all ages. 

The feeling for sound reasoning in mathematics seemed to 
have been temporarily lost. Except only -when logical rigor was 
almost unavoidable to even moderate competence, as in finite 
algorithms and combinatorial mathematics, the kind of reason- 
ing that satisfied the leading mathematicians of the so-called age 
of reason would have shocked Eudoxus and Archimedes, let 



Archimedes, no mere mathematician but a mechanist of the first 
rank, was Lagrange’s idol. It may be significant that, of all his 
own great work, Lagrange prized least highly his contributions to 
the theory of numbers, where without rigid proof for even the 
seemingly most obvious theorems there is nothing. These had 
exacted his greatest efiorts, and he doubted whether they had 
been worth their cost. There is no record of Archimedes’ having 
esteemed his practical mechanics above his mensuration of the 

Social stimuli since the death of Newton 

The transition from supcrnaturalism to rationalism in the 
exact sciences did not take place in a social vacuum. Of several 
hypotheses to account for the worship of mathcmaticizcd reason 
in the post-Newtonian age, that of economic determinism is the 
most elastic. In brief, all the work in celestial and analytic 
mechanics was occasioned by the demands of navigation and 
ballistics. However, anyone interested may search the technical 
works of Laplace and Lagrange on mechanics and find no refer- 
ence to sailing or gunnery. This docs not disprove the thesis 
that the initial impulse for the mechanics of the eighteenth 
century may have been the mercantile desirability of a reliable 
nautical almanac and the military necessity for hitting whatever 
is aimed at. It merely illustrates the verifiable fact that once a 
mathematical theory has been initiated, whatever its origin, it 
proceeds by a sort of intellectual inertia to become abstract with 
no application in sight. We shall see many instances as we 
proceed. Potential theory, for example, in spite of its mechanical 
origin, ceased long ago in those of its divisions that interest 
professional mathematicians to have any discernible relevance for 
science or technology. These abstrusities may become practically 
useful tomorrow; but only should time be reversed and evolution 
unfold inward will any application of them become their origin. 

The like holds for the calculus of variations, except that 
in this instance theology rather than science was the initial 
source in the eighteenth century. P. L. M. dc Maupertuis (1698- 
1759, French) propounded (1747) a somewhat obscure form 
of the mechanical principle of least action because he credited 
his parsimonious deity with an aversion to avoidable effort or 
other waste. This might be construed as thcologic, not economic, 
determinism unless, as some might insist, the theology was 
economically determined in the same way as the ballistics. 


Asserting that “Nature always acts by the shortest path ” 
Fermat also had gone behind observable phenomena in deriving 
his optical principle of least time. But Newton framed no super- 
fluous hypotheses in determining (1687) the surface of revolution 
offering the least resistance to motion in the direction of the 
axis through a resisting medium. Newton’s problem, were it 
proposed for the first time today, might be attributed to eco- 
nomic determinism, on account of its possible application to the 
marine torpedoes which had yet to be imagined in Newton’s 
backward time. The problem, however, has actually been attrib- 
uted to Newton’s very early advice to a young friend that the 
latter study ballistics. From these and numerous other examples 
that might be cited, it seems clear that the truth of a sociological 
theory when stretched to include all mathematics may occa- 
sionally vary inversely as its degree of elasticity. 

The fact seems to be that, if the mathematicians of the 
eighteenth century were motivated by anything less obvious 
than the desire to do mathematics and to earn their livings while 
following their inclinations, they were unaware of it. The condi- 
tions under which the great creators worked were basically 
different from those of the nineteenth and twentieth centuries. 
If an economic motivation is to be found, it might be profitably 
sought in the domestic and foreign policies of Frederick the Great 
of Prussia (1712-1786), Catherine the Great of Russia (1729- 
1796), the kings of Sardinia, Louis XVI of France (1754-1793), 
and Napoleon Bonaparte (1769-1821). The obvious demands of 
civil, naval, and military engineering made the development of 
mathematics imperative; and these rulers were clear sighted 
enough to see that the simplest way of getting mathematics 
out of a mathematician is to pay his living expenses. At various 
stages of his career, Euler was attached to the courts of Catherine 
and Frederick; Lagrange was similarly supported through the 
government- subsidized Turin Academy, a related military 
school, and later by Frederick, Louis, and Napoleon. Daniel 
Bernoulli, often called the founder of mathematical physics, was 
employed by Catherine. Monge and Laplace were employees o 
successive French governments in various capacities . from 
military engineering and the training of government engineers 
to affairs of state, and likewise for Fourier and a number of e> s 
distinguished mathematicians. But before the Napoleonic era, 
once these men had advised their employers on the technic^ 
questions, usually simple, proposed to them, they were free o 



spend their working time as they chose. Consequently, an enor- 
mous amount of mathematics having no detectable application 
was created. That much of it proved of practical value years or 
decades later does not alter the fact that its motivation was not 
economic. All this was printed with the rest largely at public 
expense in the proceedings of government-subsidized academies. 
Up to the Napoleonic era, the learned societies were the 
most important agencies for the publication of research in 

With the eruption of French TLiberty, Equal it}', Fraternity’ 
in 17S9, a rapid democratization of mathematical research 
began. Under Napoleon, the leading French mathematicians 
earned part of their keep by helping to train civil and military 
engineers at the Ecole Polytechnique. Others at the Ecole Nor- 
male Supericure taught prospective teachers. Again the major 
part of the new mathematics produced by the leaders was of no 
immediate practical value, nor was it undertaken with a view to 
possible applications. Napoleon no doubt was partly responsible 
for this liberality. Provided the schools supplied him with a 
steady flow of competent civil servants and expert engineers to 
fill the rather frequent vacancies that might be anticipated in a 
militaristic regime, he was content. Some of the mathematicians 
compiled excellent texts for the students; others carefully pre- 
pared their few lectures a week; and nearly all did research in 
their ample spare time. The situation was not unlike that in a 
few of the more enlightened European and American universities 
of the twentieth century. 

The next and last marked change in the social status of 
mathematics and mathematicians dates from the decade 1816-26 
following the end of the Napoleonic era in ISIS. The universities 
and technical schools increased rapidly in importance as centers 
of mathematical research; the learned societies were no longer 
financially able to cope with the torrent of new mathematics 
that gushed from a hundred sources; and, most important of 
all, what was everybody’s business became nobody’s business. 

Whatever else may be said for democracy, it has consist- 
ently fostered the individual freedom of mathematicians. No 
mathematician in a democracy is constrained to create mathe- 
matics at public expense. All may earn their livings as they 
please and find what time they can to advance their hobby. 
Nor are mathematicians as a class debauched by having their 
researches printed at public expense, although such work can 


be neither patented nor copyrighted and may be used by an Vo 
without payment of any kind. In the United States the more 
liberal universities subsidize the publications of their staffs 
It is admitted by the majority of educated persons that a 
technological and scientific civilization without mathematics is 
an impossibility. Apparently the most efficient way of getting 
the necessary job done is to leave it to the initiative of indi- 
viduals on their own time after a more or less exhausting day’s 
work. Perhaps rather unexpectedly, the result has been a vastly 
increased output of mathematics since__the close of the Napo- 
leonic era over all preceding history. 

The strongest stimuli have been the constantly growing 
demand for scientific and technological instruction to keep pace 
•with the rest of modern civilization, and the vast expansion of 
media for mathematical publication since the German engineer 
A. L. Crelle (1780-1855) in 1826 subsidized the first high-grade 
mathematical periodical. In 1940 there were about 280 such 
periodicals 1 devoted wholly or in part to the publication of re- 
search in mathematics, and the pressure on existing outlets was 
steadily increasing. Few if any of these publications would sur- 
vive for two months if forced to pay their way in a competitire 
society. They are supported by the whole mathematical frater- 
nity without distinction as to race, nationality, or creed. The 
average mathematician subscribes to as many of them as he 
can afford, even if but very few of the severely technical articles 

printed in a year are within his comprehension. 

It might be difficult to account for this curious phenomenon 
on strictly economic grounds. A majority of the subscribers 
do no research themselves; so it cannot be polite hints from 
superiors that they shed luster or notoriety on their employers 
by getting their names into print that account for the altruistic 
subscriptions. Nor will a mathematically literate reader scanning 
the abstracts of current research agree that any considerable 
percentage of the articles printed by the hundreds every mont 
were inspired by economic or other practical needs. _ 

Several thousand 2 periodicals devoted to engineering an 
other exact sciences take care of the immediately P ractica 
applications of mathematics. But these are not the journals 
to which mathematicians subscribe. Many of. these ©or 
practical journals pay their way in the competitive mar 
This, possibly, is the nub of the distinction between p 

and applied mathematics. 



In following the influence of science on mathematics, it 
should be remembered that mathematics and its applications are 
different things. A treatise or a monograph on mathematical 
physics, for example, may be a mass of formulas and equations 
from beginning to end, and yet make no contribution whatever to 
mathematics. If the general fact is not ob%'ious, its extreme 
cases, as in bookkeeping or the calculation of characteristic 
functions in quantum mechanics, may illuminate the distinction. 
As a further aid to comprehension, it is a fair guess that out of a 
hundred thousand persons picked at random on the streets of 
New York, or Chicago, or London, or Paris, or Moscow, or 
Tokyo, not one would know the name of the man whom profes- 
sional mathematicians almost unanimously considered was the 
foremost member of their guild since about 1912. He died in 
1943, inactive; but his fame is secure even if the average man 
in the street (or in cultured society) is never likely to hear of 
him. Of the random hundred thousand, many would instantly 
name a theoretical physicist who deeply resents being called a 


From Mechanics to Generalized 


Of all the exact sciences, mechanics, the simplest, has prob- 
ably been the most influential in the development of modern 
mathematics. The amount of known mathematics applied to a 
science is no measure of the importance of that science in mathe- 
matical evolution; it is the new mathematics inspired by a partic- 
ular science alone that weighs. 

Thus in the first twenty years (1925-45) of its existence, 
quantum mechanics used an enormous amount of mathematics, 
from special functions to modern algebra, but did not suggest any 
essentially new mathematics. 1 General relativity, on the other 
hand, drawing less heavily on mathematics, was directly re- 
sponsible for the direction taken by differential geometry 
about 1920. This newer geometry might have been developed 
almost forty years earlier. All the necessary technique was 
available; but it was not until the successes of relativity showed 
that Riemannian space and the tensor calculus were of more 
than mathematical interest that differential geometers noticed 
what they had been overlooking. 

Before considering the mechanical origins of certain parts 
of analysis, we shall give brief summaries of the relevant progress 
in mechanics in the eighteenth and nineteenth centuries partly 
responsible for the mathematics. 

The search for variational principles 

One purpose of the eighteenth-century mechanists was the 
invention of principles from which the mechanics of Galileo an 



Newton could be deduced, and the development of mathematics 
adequate for the deduction. The main mathematical outgrowths 
were the calculus of variations; a vast theory of differential 
equations; a heterogeneous collection of special functions; the 
beginnings of the theory of line, surface, and volume integrals; 
more than a hint of ^-dimensional space; 5 the origins of potential 
theory; and certain basic results in what subscaucntly became 
the theory of functions of a complex variable. 

The first comprehensive principle of post-Newtonian me- 
chanics was D’Alembert’s, published in his Traitc de dynamique 
(1743): the internal actions and reactions of any system of rigid 
bodies in motion arc in equilibrium. Or, as often expressed: in 
a dynamical system the reversed effective forces and the im- 
pressed forces arc in equilibrium. 

Supplementing Newton’s principles of the conservation of 
momentum and of the center of mass, Euler and Daniel Ber- 
noulli (1700-1782) independently stated (1746) the principle of 
conservation of areas. All of these foreshadowed the concept of 

Euler’s Mcchanica, sive viotus scientia analyticc exposita 
(1736), was a halfway house between the purely geometrical and 
synthetic mechanics of Newton’s Principia (1687) and La- 
grange’s Mechaniquc analytique (1788). Euler sought to replace 
synthetic methods by analysis, and %vas largely successful. 
Visual geometric intuition, however, was still used, as in the 
resolution into tangential and normal components in curvilinear 
motion. Much of the scrappy geometry of curves and surfaces 
embellishing antiquated texts on the calculus under the com- 
prehensive rubric ‘geometrical applications’ originated in this 
way. Possibly mechanics is also partly responsible for classical 
differential geometry and the intrinsic geometry of curves. 

Taking a considerable step toward a general method, C. 
Maclaurin (1698-1746, Scotch), in his Complete system of 
fluxions , 1742, advanced beyond Euler by using three fixed axes 
for the resolution of forces. The advantage of Maclaurin’s 
procedure over Euler’s is comparable to Descartes’ use of one 
coordinate system to display any number of curves. But the 
mathematical formulation of each type of problem still required 
special devices. 

Introducing his generalized coordinates, Lagrange in 1760 
turned away from mere ingenuity, and started toward the 
general equations of motion on which he based his analytic 


mechanics of 1788. The equations of motion for a holonomic 
dynamical system were then obtained in a form adaptable to the 
special coordinates most convenient for particular problems. 
The distinction between holonomic and non-holonomic systems 
may serve to illustrate certain concepts of mechanics which 
were partly responsible for the calculus of variations in its 
earlier form. 

To exhibit these, it will be necessary to use the “arbitrary 
infinitesimal displacements” in terms of which applied mathema- 
ticians frequently think. It is not easy to give a mathematically 
sound treatment of the related variational operator 3 6 and 
the infinitesimal displacements that lead to useful results before 
they finally disappear from the calculations; and one rather 
extreme school advocates abandoning all pretense of deriving 
the dynamical equations by such means. It would be closer to 
modern science to state the general equations of motion as 
postulates, the sole function of these equations being the 
mathematical statement of dynamical problems, for which the 
equations themselves are adequate. Their deduction by more or 
less mystical reasoning dating from ancient Greece and the 
Middle Ages is of purely historical interest, contributing 
nothing to understanding or utility. However, as these vestiges of 
an older mode of thought are still helpful to the majority of 
applied mathematicians, we shall follow tradition even where it 
is now asserted by rigorists to be unsound, and by modernists in 
theoretical physics to be meaningless. 4 

The configuration of a dynamical system, regarded as being 
composed of material particles subject to constraints (as that all 
the particles move only on given surfaces) and geometrical 
conditions (as that the distance between any two given points 
of a rigid body is constant), is specified at time t by n coordinates 
<7i, qi, q n , where n is finite. If, for example, the Cartesian 

coordinates of the rth particle at time t are x r , y r , %r, the system 
is specified by 3m equations 

= Mqi, . . . ,q n ), y r = gXqu • • • > ?«)> 

z, = h r (q u . . . , q n ); r = 1, ' 

Let each of the generalized coordinates qu ••• !?» receive an 
arbitrary infinitesimal increment; say the increments are 
Sq i, . . . , 8q n . There is not necessarily a physically possi. e 
displacement of the system corresponding to 8q i, • • • > oQm 1 
there is, the system is called holonomic; if not, non-holonomic. 


A holonomic system specified by ?i, . . . , q n is said to have n 
degrees of freedom. 

Lagrange’s equations for a system with n degrees of freedom, 
for which a potential function exists, 5 can now be stated. 

The derivatives of . . . , q„ with respect to t being de- 
noted by q i, . . . , q n , and the difference T — V between the 
kinetic energy T and the potential energy V of the system by 
L, the equations of motion are 

d_ (0L\ 

dt \dq r J 

T *— 1 , « ■ * , 

L is called the Lagrangian function, or the kinetic potential, of 
the system. 

The point of historical interest here is in the ‘small dis- 
placement’ (oqt, . . . ,8q„)oi(qi, . . . , q n ). By a route which 
need not be retraced, its interest being mechanical rather than 
mathematical, (5qi, . . . , 8q r ) descended from the virtual 
displacements used by Stevinus, Descartes, and others in statics. 
Virtual displacements appear fully matured in the principle of 
virtual work, which was one of the clues followed to an analytic 
mechanics by Euler and Lagrange. An extremely liberal inter- 
pretation of ancient and medieval mechanical speculations has 
enabled some scholars to detect elusive hints of virtual work all 
the way back to the Greek philosophers. Virtual displacements, 
virtual velocities, and virtual work arc obviously in the general 
direction of a calculus of variations. The next major advance 
in analytic mechanics was in the same direction. It finally 
reduced the mathematics of statics and dynamics to a topic in 
the classical calculus of variations. 

The statement of several mechanical theorems of the seven- 
teenth and eighteenth centuries had suggested to Euler that all 
natural phenomena present extrema, and that physical princi- 
ples, including those of mechanics, should be expressible in 
terms of maxima and minima. For example, Huygens had shown 
that Fermat's optical principle of least time holds for media 
whose index of refraction varies continuously from point to 
point; James and John Bernoulli had found the catenary as the 
arc of fixed length passing through two fixed points and having 
the lowest center of gravity; John Bernoulli’s problem (1696) of 
finding the curve 5 of quickest descent under gravity from one 
fixed point to another in a vertical plane had been correctly 


solved by L’Hospital, Leibniz, Newton, James Bernoulli, and 
John himself; and last, Euler had sought a function whose 
variation equated to zero would yield the differential equations 
of dynamics. For a single particle, the ingenious Euler observed 
that if the velocity v is given as a function of the coordinates 
of the particle, the desired equations are obtained by minimiz- 
ing J» ds, where ds is an element of the path in which the particle 
moves. Otherwise expressed, the equations of motion are found 
on performing the variation 8 jv ds and equating the result to 
zero. In this way Euler was led to minimize definite integrals 
in which the integrand is of the form f(x, y) (1 -j- 
y' = dy/dx, and the integration is with respect to x. 

It is to be noted that Euler, guided by intuition, sought 
minima to express natural ‘laws,’ possibly because he was of 
the same pietistic cast of mind as Maupertuis. But Jacobi in his 
lectures on dynamics (edited, 1866), produced an almost 
trivial mechanical problem in which the action is a maximum. 
It is therefore customary to speak of stationary values of definite 
integrals for the expression of physical laws, rather than to 
prejudge the issue by expecting a least, a definite integral whose 
variation vanishes being said to represent a stationary value. 
The vanishing of the variation is insufficient to secure either a 
maximum or a minimum, although in many physical situations 
it is otherwise evident that a definite one of these must occur, 
and it is seldom necessary to proceed further. But modern 
science does occasionally demand more than shrewd guessing 
regarding extrema. Thus, in 1939, R. C. Tolman encountered a 
capital problem in astrophysics for which scientific intuition 
seemed insufficient, and for which the more refined techniques 
of the calculus of variations were at least helpful. 

Euler’s project was completed in 1834-5 by Hamilton, who 
showed that the dynamical equations are obtainable from a 
simple stationary principle, which, for a conservative system, is 

8 j^ 1 L dt = 0, where to, t\ are the initial and final times for the 

passage of a dynamical system with the kinetic potential L 
from one given configuration to another. A verbal equivalent is 
as follows. Of all possible motions by which a dynamical system 
may pass in a given time from one given configuration to 
another, the actual motion will be that for which the average 
value of the kinetic potential is stationary. The analytic equiva- 
lent of ‘possible’ is the process of variation, which evolved, at 


least in mechanics, from virtual displacements. Hamilton also 
gave a variational principle for non-conservative systems. 

The variational principles of mechanics are far from ex- 
hausted by those noted. Thus Gauss (1829) reformulated and 
generalized D’Alembert’s principle in his own of least restraint; 
and Hamilton’s principle, also that of least action, were extended 
in the 1890’s to non-holonomic systems. In his Die Prinzipicn dcr 
Mecha nil: (1894), H. R. Hertz (1857-1894, German) reworked 
the subject in terms of geometrical imagery. With the develop- 
ment of metric differential geometry for space of n dimensions 
in the second half of the nineteenth century, it was apparently 
inevitable that the stationary principles of dynamics should be 
rephrased in the language of geodesies. But here, as elsewhere 
after Lagrange, whatever gain there may have been was scien- 
tific rather than mathematical, new techniques in geometry and 
analysis suggesting reformulations of mechanics. Lie’s theory of 
contact transformations also was elaborated for its mathematical 
interest long before it was applied (1889) to unify the differential 
equations of dynamical systems, although the connection be- 
tween dynamics and contact transformations was implicit in 
Hamilton’s work of 1834-5. The like appears to be true of 
Poincare’s integral invariants (1890), also of the topological 
methods first applied to dynamics by him and since extensively 
developed by a prolific company of pure mathematicians. 7 
In the last instances, however, outstanding problems of dynami- 
cal astronomy, such as that of three bodies mutually attracting 
one another according to Newtonian gravitation, 8 were the 
ultimate source of the mathematics. 3 

Enough has been given to suggest that mechanics was an 
important source of the calculus of variations. Before passing 
on to a summary account of the development of that branch of 
analysis, we may glance at the scientific significance of varia- 
tional (stationary) principles in general. 3 

Competent opinion is sharply divided. In the tradition of 
Maupertuis, Euler, and their predecessors, one side professes 
to see cosmic profundities in the derivation of Lagrange’s 
equations from a variational principle. The profundities are no 
longer theological, as in the eighteenth century, but concern 
unapprehended necessities of the physical universe. Conse- 
quently it is claimed that a real but not wholly understood 
scientific advance is made when the differential equations of a 
physical theory arc shown to be obtainable from a variational 


principle, as was done, for example, quite early by Hilbert in 
general relativity. 

The other side holds that variational principles are incapable 
of adding to science anything not already known in a form better 
adapted to calculation. For this side, any variational principle 
in science is at most only a concise restatement of more or less 
ancient history which might become useful, because it is easily 
remembered, should all working scientists suddenly forget the 
mathematics they actually use to obtain results that can be 
checked against observation. This side further asserts that 
the reformulation of a science in terms of a variational principle 
is what the average modern pure mathematician does when he 
attempts to contribute to science, a task for which he is not 

Less immoderate observers occupy a middle position, 
pointing out that if physicists had scrutinized the duality in 
Hamilton’s optics and dynamics, in which the principles of 
least time and least action were shown to be interrelated, they 
might have come upon de Broglie’s waves and Schrodinger’s 
wave mechanics about ninety years earlier than they did. But 
this belongs to the elusive metaphysics of might-have-been, and 
cannot be considered as a promising suggestion for the future. 

There remain, then, only the two extremes with no tenable 
ground between them. Their respective creeds reflect those of 
the corresponding sides in mathematics. The disciples of 
Maupertuis would favor the vision of mathematics as eternally 
existing and necessary truth; their opponents would see mathe- 
matics only as a humanly created language adapted to definite 
ends prescribed by human beings. It is a matter of individual 
preference which of these, if either, is considered the worthier. 

Functions as variables 

Problems of maxima and minima in the differential calculus 
seek those values of the independent variables for which a given 
function of them assumes a greatest or a least value. The varia- 
bles represent real numbers. 

In the calculus of variations it is required to determine 
one or more unknown functions so that a given definite integral 
involving those functions shall assume greatest or least values. 
The variables here are functions. As the simplest example, it 
is required to find the shortest arc joining two fixed points 
(*!, (x 2 , yi )* All the infinity of arcs y = /(*)> *1 = x = Xl ’ 


joining the two points satisfy the end conditions y\ ~ f(x i), 
y, =/(.Vi). The shortest will be that one (or those, should 

there be several) of this infinity which makes f: (1 + yydx, 

where y' denotes dy/dx, a minimum. 

The solution here is intuitively evident, and therefore open 
to suspicion. It is also ‘obvious* that the plane closed arc of 
given length enclosing the maximum area is a circle; and like- 
wise for a sphere as the surface of given area enclosing the 
maximum volume. But ‘obvious’ as the last two theorems are, 
the Greek geometers 10 attempted to prove them by elementary 
geometry. Discounting the legend of Queen Dido and her bull’s 
hide, 11 we have in the first of these isoperimetric problems of the 
ancients the earliest concerning maxima and minima that 
have been rigorously solved only by the calculus of variations. 
Solutions were delayed till the second half of the nineteenth 

The simplest mechanical problems of the seventeenth and 
eighteenth centuries involving minima, such as that of the 
brachistochrone, 10 transcend intuition but are still within range 
of ingenious geometry. John Bernoulli in 1697 solved the 
brachistochrone problem elegantly by special devices, using 
nothing more advanced than an integration. His brother James 
far surpassed him in the same year with an inelegant but more 
general method of solution applicable to a wide class of problems. 
James Bernoulli’s 13 signal merit was his recognition that the 
problem of selecting from an infinity of curves one having a 
given maximum or minimum property was of a novel genus, 
not amenable to the differential calculus and demanding the 
invention of new methods. This was the mathematical origin 
of the calculus of variations. 

The development of the subject is detailed and intricate, 
especially in the recent period; and we can give only the briefest 
summary sufficient to indicate the part played by the calculus 
of variations in the development of modern analysis. Intrin- 
sically more difficult than some of the other major divisions 
of analysis, such as the classical theory of functions, the cal- 
culus of variations has attracted relatively fewer specialists. 
But those who have made it their chief concern seem to have 
been embarrassingly prolific. 

For our purposes here, the points of greatest interest are 
the early emergence of the new calculus as an independent 


department of analysis concerned only incidentally with the 
mechanical and geometrical problems in which it originated 
and the progression toward a theory of functions of a non- 
denumerable infinity of variables. The calculus of variations 
itself is not concerned with the last; but the minimizing arcs 
of the theory suggest an infinity of variables in two respects. 
An extremal (a minimizing or maximizing arc) subject to given 
end conditions is one of an infinity of variable arcs; the arc 
itself is an infinite set of points. These hints appear to have been 
partly responsible for the theory of functions of lines (‘func- 
tionals’) and the geometry of spaces of an infinity of dimensions. 

There were roughly six stages in the development. The first 
extended from the last decade of the seventeenth century to 
about 1740, and is typified by the work of the Bernoullis. The 
second opened in 1736 with Euler’s 14 differential equation 
giving a necessary condition for a minimizing curve. In 1744 
Euler gave a systematic exposition of his method, with needed 

Abandoning Euler’s semi-geometrical attack, Lagrange 
(1762, 1770) passed to the third stage with an analytic method 
which furnished the differential equations of the minimizing 
curves. He introduced the variational operator 6 and developed 
its algorithm, greatly simplifying and extending most of the 
work of his predecessors. With Lagrange, the calculus of varia- 
tions become an autonomous division of analysis. 

The fourth stage, 1786-1837, began with Legendre, who 
investigated the second variation of an integral to find criteria 
for distinguishing between maxima and minima. This was 
analogous to the use of the second derivative for the like purpose 
in problems of maxima and minima solvable by the differential 
calculus. Legendre’s criteria were inconclusive; Jacobi (1837) 
gave a critical evaluation of Legendre’s analysis, discussing 
when it would lead to the desired end and when it would not. 
Jacobi was thus led to his geometrical interpretation of his 
own criterion in terms of the conjugate point 16 which he defined. 

For about forty years — a long time in modern mathematics 
after Jacobi’s advance, there was no significant progress. But 
analysis in the meantime was undergoing a basic revision. 
Weierstrass, “the father of modern analysis,” was transforming 
the mathematics of continuity into a rigid logical system bearing 
but little resemblance to the intuitive analysis of most of his 
predecessors. His lectures 16 of 1879 at the University of Berlin 


on the calculus of variations mark the beginning of the fifth 
stage. With almost Gaussian indifference to fame, Weierstrass 
contented himself with lecturing on his revision of the theory; 
and although his work was not printed in his lifetime, it pro- 
foundly influenced the entire future development through the 
research and teaching of his students. Of the latter, one in 
particular may be mentioned here, O. Bolza (1857-1942, Ger- 
man) whose lectures over several years at the University of 
Chicago were responsible for the highly productive American 
school in the modern calculus of variations. Bolza attended 
Weierstrass’ lectures of 1879. 

In addition to rigorizing the entire subject as it ezisted 
in his time, Weierstrass made extensive additions of his own. 
To him are due a new sufficiency condition and the first accept- 
able sufficiency proofs, for which he invented his fields of 
extremals. For the geometrical interpretation of his analysis 
he used the parametric equations of curves, with a consequent 
gain in generality. This step may have been suggested by the like 
in differentia! geometry, which had been current since Gauss 
(1827) made extensive use of it in his study of surfaces. It goes 
back even farther, to Lagrange’s generalized coordinates in 
dynamics as functions of the time; but Weierstrass was the 
first to apply it to the calculus of variations. 

The Wcierstrassian period lasted into the twentieth century. 
Its standards of rigor persisted as problems of increasing gen- 
erality were attacked by modern analysis. There were notable 
applications to differential geometry in the 1890’s, as in the 
work of G. Darboux on geodesics, later generalized by several 
other men. The sixth stage dates from 1899-1900, beginning 
with Hilbert's proof of his differentiability condition for a 
minimizing arc, assuring in many problems the existence of an 
extremal, and his exploitation of the invariant integral since 
named after him. Finally, in 1921-3, L. Toncili, (1885-, Italian) 
opened a new chapter, proceeding from PlilberTs work to a 
revision, concerned principally with existence theorems, of the 
entire calculus of variations. 

The individuals whose names have been mentioned are not, 
of course, the only men whose labors have created the calculus of 
variations; nor arc the few advances noted an adequate measure 
of the rich complexity of this subtle division of analysis. Scores 
of men have contributed hundreds of theorems, until here as 
cbewhere in modern mathematics what was a narrow specialty 


in the early nineteenth century began in the twentieth to split 
into still narrower specialties, each with its assiduous corps cf 
cultivators. Only an expert who has devoted his working life 
to the subject can take in the •whole of it or estimate the vitality 
of its several subdivisions. The same is true for any major 
department of modern mathematics; and it may be taken for 
granted that any short report on a particular topic can indicate 
only a few of the salient characteristics. 

The same general features as in the rest of recent mathe- 
matics stand out in the development of the calculus of varia- 
tions, with one possible difference: some of the most difficult 
problems appeared early and were partly solved by ingenious 
men who could not possibly have realized how hard the problems 
were. Otherwise the progress from special problems to others 
more inclusive or less restricted followed the familiar pattern 
of generalization with increasing rigor. Instead of problems 
concerning arcs with fixed end-points, problems with variable 
end-points were considered, the earliest being James Bernoulli’s 
(1697) of the curve of quickest descent under gravity from a 
fixed point to a fixed vertical straight line, a problem with one 
variable end-point. Generalization in another direction pro- 
ceeded by modification of the integrand in the definite integral 
to be minimized. A third type of generalization combined the 
first two, superimposing generalized end-conditions on the 
function to be minimized. A far-reaching generalization of 
this kind was 0. Bolza’s of 1913, which included several famous 
problems as special cases, among them Lagrange’s of 1770 
and A. Mayer’s of 1878. Since about 1920 the greatest activity 
in this direction has been in the United States; indeed, shortly 
after 1900 the calculus of variations became a favorite field of 
research with American mathematicians, of whom G. A. Bliss 
(1876-) and his numerous pupils, and M. Morse (1892-) were 
particularly active. 

Although we cannot discuss special problems, one may be 
mentioned for its historic interest. J. Plateau’s (1801-1883, 
Belgian) problem (1873), first proposed by Lagrange, to deter- 
mine the surface of least area with a given boundary is solved 
physically by the soap film which spans a wire model of the 
boundary. A complete mathematical solution was given only w 
1931 by J. Douglas (1897-, U.S.A.). 

In the calculus of variations we have seen the first extensively 
developed department of analysis in which functions of variable* 


other than those discussed in the ordinary calculus are con- 
sidered. This long step forward was to prove of more than local 
significance. Much of the analysis of the twentieth century is 
concerned with functions of generalized variables, and with the 
corresponding abstract spaces created to provide the appro- 
priate geometrical description of the analysis. Looking back 
on the analysis of the eighteenth and nineteenth centuries, 
we observe many trends toward what has been called general 
analysis. Enough of these generalizations to indicate the develop- 
ment of some kind of general analysis, and the need for it, will 
be described in later chapters. 


From Applications to 

In the progression toward general theories of analysis, the 
special functions devised in the eighteenth and early nineteenth 
centuries for the solution of problems in dynamical astronomy 
and mathematical physics played a dominant part in determin- 
ing the course of modern analysis. From the historical record, 
it seems incredible that some of the special functions, for example 
those of Bessel and E. Mathieu (1835-1890, French), would 
ever have seriously engaged the attention of mathematicians 
had it not been for the initial impulse from science. But not all 
of the most extensively investigated functions can be credited 
exclusively to scientific necessity. Thus the multiply periodic 
functions developed inevitably from the straightforward evolu- 
tion of the integral calculus. A few typical cases 1 will suffice to 
illustrate the general trends. 

A central problem of applied mathematics 

The hardest thing in any applied mathematics is to strip a 
scientific or technological problem of enough details, and no 
more, to bring it within the capabilities of skilled mathematicians 
and still preserve sufficient of the actual problem to make 
the solution not utterly irrelevant for practical applications. 
Observation presents us with no motion immune to friction,, 
and no incompressible fluid; yet the classical hydrodynamics' 
of incompressible fluids without viscosity has had many applica- 
tions. The all-important problem of deciding what concepts 
to be made central in the mathematical description of natural 


phenomena is of a like character, and requires the same rare 



combination of scientific insight and mathematical tact for its 
successful solution. Velocity in kinematics, entropy in thermo- 
dynamics, also force, action, and energy in dynamics illustrate 
the point. A more recent instance is correlation in the statistical 

The great mathematicians of the eighteenth century ex- 
celled in this most difficult field. The distinction between pure 
and applied mathematicians did not exist, nor was it necessary, 
when the Bcrnoullis, Euler, d’Alembert, Clairaut, Laplace, 
Legendre, and Mongc were at their best. It was largely due to 
their colossal output of both pure and applied mathematics that 
it became humanly impossible by the middle of the nineteenth 
century for a man to attain the first rank as a scientist and as a 

As we look back on all this seething activity, we observe 
the hesitant beginnings of theories which were to occupy thou- 
sands of industrious mathematicians from the early nineteenth 
century to well within the twentieth. Following one of these 
along the clue provided by the Bessel functions, which are 
among the most useful functions in mathematical physics, we 
shall be led to a central problem of applied mathematics. This 
problem generated numerous special functions; and from these 
in turn some of the major divisions of modern mathematical 
analysis evolved. 

Investigating the oscillations of heavy chains, Daniel Ber- 
noulli* (1700-1782, Swiss) in 1732 encountered the function 
later called a Bessel coefficient of order zero. Bessel coefficients 
of order had appeared earlier in a problem of James Bernoulli’s 4 
(1654-1705). The vibrations of a stretched membrane led Euler 5 
in 1764 to more general Bessel coefficients, and seven years later 
Lagrange encountered the same functions in elliptic motion. In 
1824, the mathematical astronomer F. W. Bessel (1784-1846, 
German), needing these functions in his investigation of a 
perturbative function in dynamical astronomy, developed sev- 
eral of their more useful properties. Thereafter, the Bessel 
coefficients and their immediate extensions, the Bessel functions, 
appeared in physical science almost as frequently as the circular 
functions, and chiefly for the reasons indicated next. What 
follows is relevant for our entire subsequent discussion of the 
influence of the physical sciences on mathematics. 

I he advantages of special coordinate systems adapted to 
specific problems were familiar to geometers before a similar 


specialization in applied mathematics was recognized as an 
ultimate source of the indispensable special functions, such as 
Bessel’s, of astronomy and physics. In discussing physical 
situations involving symmetry about a straight line, for example, 
it is convenient, indeed almost mandatory, to use cylindrical 
coordinates (r, <$ , z), just as it is to use spherical coordinates 
(r, 0, <p) where there is symmetry about a point. When Laplace’s 
equation 0 V 2 « = 0 is transformed from rectangular to cylindrical 
coordinates the variables are separable, and Bessel’s differential 
equation drops out as that which r must satisfy. The same 
equation appears similarly in the transformation to spherical 
coordinates of the equation 7 \V 2 u = dv/dt, to which Fourier was 
led in his analysis of heat conduction. A typical problem of 
great generality connected with this equation may serve to 
illustrate the central problem of applied mathematics which 
we have in view, that of boundary-values, in which special 
functions, such as Bessel’s, are only details of calculation. 

The typical problem is to find a solution of Fourier’s equa- 
tion subject to the following conditions. At each point (x, y, z) 
of the interior of a homogeneous isotropic solid, the temperature 
v (satisfying the equation) is to be a continuous function of 
x, y, z, t, having continuous first and second partial derivatives 
with respect to x, y, z, and having dv/dt continuous. The tem- 
perature v throughout the body at the initial time t = 0 is to be 
given by v — fix, y, z), where / is an arbitrary continuous 
function; and the solution v, obtained as a function of x, y, s, t, 
must be such that its limit as t approaches zero is f{jx, y, z). It 
may be assumed that if two bodies of different conductivities 
are separated by a common boundary, the temperatures of the 
bodies at any point of the boundary are the same. 8 The problem 
is easily modified to take account of radiation into a surrounding 
atmosphere: the loss of heat per unit area of the boundary is to be 
proportional to the difference in temperature between the surface 
and the atmosphere, in accordance with an empirically estab- 
lished law of cooling. Finally, the temperature at any point 
(x, y, z) of the boundary at time f may be prescribed as a given 
continuous function F(x, y, z, /). The solution v of Fourier s 
equation satisfying these conditions is unique. Special problems 
of this type leading to Bessel functions are the flow of heat m a 
circular cylinder or in a sphere whose surface is maintained at 
zero temperature. 


This typical problem is a specimen of boundary-value prob- 
lems, in which it is required to construct that solution of a given 
differential equation, ordinary or partial, that fits prescribed 
initial conditions. If properly posed, the problem has a unique 
solution; but, as will appear, it is not always obvious that all 
the conditions of a given situation have been included in the 
mathematical formulation, or that, if included, they are analyti- 
cally compatible. The theory of such problems is coextensive 
with a vast tract of mathematical physics, and has given rise 
to equally extensive tracts of pure mathematics connected, if at 
all, only remotely with practical or scientific applications. 

Many of the classical boundary-value problems in mathe- 
matical physics lead to analogues of Fourier’s project of expand- 
ing an ‘arbitrary’ function /(.v) in a trigonometric series in the 


.v-interval — tt tom, say /(.*) = -Uj 0 + X cos nx "b s * n nx )> 

n *- 1 

where the coefficients a 0} a n , b n arc to be determined. Under 
certain restrictions, the coefficients arc given by 

- f' r f(y) cos my dy, 

7rb a = J_ r f(y) sin my dy, (m £ 0). 

The point to be noted here is that/(x) is expanded in terms of 
the solutions cos mx, sin mx, of the ordinary differential equation 
dhi/dx- -f- vru = 0. 

A central problem of mathematical physics is a generaliza- 
tion of this: it is required to expand a suitably restricted func- 


lion f(x) in a scries of the form c D + X f n$n(*), where the 

n — 1 

functions <j>n(x) arc solutions of a given ordinary linear differ- 
ential equation. The possibility of the expansion being assumed, 
the problem amounts to calculating the coefficients Co, Cj, c«, 
. . . . The conditions under which the scries converges must 
then be determined, if the expansion is to be usable. 

It seems conservative to say that the majority of those 
special functions which have been most exhaustively investigated 
since the early eighteenth century entered mathematics in this 
way through the differentia! equations of astronomy and physics. 
Although many of them, like the Bessel coefficients, appeared 
first in a rather haphazard manner in mechanical problems of 



the early eighteenth century, their wider significance began to 
emerge only with the problem of separation of variables in the 
partial differential equations of potential theory and other 
departments of mathematical physics. This led directly to the 
expansion problem just described, and to the modern theory 
of boundary-value problems which furnishes the desired coeffi- 
cients and justifies the expansions. This phase of the general 
development of analysis will recur frequently as we proceed. 

Once the special functions had fulfilled the more immediate 
scientific purposes for which they had been invented, they were 
exploited by numerous analysts whose interests were purely 
mathematical. Scientific applications were not even remotely 
envisaged in the continually refined generalizations 9 of the 
analysis that had sufficed for physical problems. From one 
point of view, this rapid transition from the immediately applica- 
ble to the abstract with no application in sight seems only 
natural and typical of the general progress of mathematics. 
Admitting that the development is typical, we may nevertheless 
question its curiously fortuitous character. The Bessel functions 
may serve once more as an illustration. 

It has often been said by analysts with a taste for elegance 
that no mathematician left to his own devices would ever have 
dreamed of inventing anything so uncouth mathematically as the 
Bessel] functions; or, if by cha'nce he had imagined such things 
in a nightmare, he would have done his utmost to forget them 
on coming to his senses. Such elegancies as these functions 
may exhibit in the refinements of twentieth-century analysis, 
as in the theory of various transforms or in applications to the 
theory of numbers, were unimaginable to the eighteenth-century 
mathematicians, whose motives in investigating special cases 
of the functions were wholly scientific or practical. Whether 
defined by infinite series or by a differential equation, there 
was nothing about the Bessel coefficients as first presented to 
suggest that they and their generalizations might repay ex- 
haustive investigation on their own account. 

The like holds for many of the other special functions con- 
ceived in science and born into technology, for example the 
Mathieu 10 functions, introduced (1868) to analyze the vibra- 
tions of an elliptic membrane. All the intricate analysis that 
developed from these scientific origins seems strangely parasitic 
and accidental to those who believe that mathematics evolves 
in response to the dictates of an indwelling and eternal necessity. 


To these, some of the most highly prized acquisitions of modern 
mathematics arc mere by-products of chance. There is, they 
maintain, neither reason nor necessity in the selection of what 
particular things arc to be developed; and almost any choice 
other than that actually made would produce results equally 
pleasing to a mathematician. So say the practical realists, who 
also occasionally take mathematicians to task for fleeing to 
abstractions when seemingly more fertile fields await cultivation. 

Against this opinion, it is contended that mathematicians 
as a class prefer the problems of pure mathematics to those of 
applied because to do so is merely to follow the line of least 
resistance. Centuries of trial and error have shown in what 
directions advances may be anticipated for a moderate expendi- 
ture of thought; and the same process of elimination has simul- 
taneously suggested the means of progress. This appears to 
be the basic reason for the phenomenal popularity of abstract 
algebra, abstract spaces, and general analysis in the twentieth 
century. Of the endless variations implicit in the syntax of 
mathematics at any stage of its development, those following 
most closely what has already been explicated are usually 
selected for further elaboration. We shall see a striking and 
historically important instance of this presently, when we 
consider elliptic functions. 

Mathematics and scientific intuition 

It is ‘intuitively evident’ that electricity applied to a 
bounded conductor will reach a definite and unique distribution 
when the conductor is fully charged and no more electricity flows 
onto it. But it is not evident mathematically. Intuition in 
mathematics frequently acts as a decoy to credulity. It was so 
in the evolution of the calculus, and it is so here. The physically 
evident assertion about the conductor may in fact conceal an 
ineradicable incompatibility. It may be too sweeping a generali- 
zation from crude observations. 

The difficulties begin when intuitive notions of a boundary 
are made precise. Arc Pcano’s area-filling curves, for example, 
to be admitted as boundaries ? When these and similar conditions 
have been agreed upon, intuition has departed. Let intuition 
state offhand what will be the distribution of electricity on 
a one-sided conductor, or on a body like a cactus pad with 
spines tapering off exponentially to infinity. But, it may be 
legitimately objected, neither of these abnormalities ever ap- 


pears in nature or technology. Granting this, we are left with 
the severely practical problems of deciding which conductors sub- 
mit to mathematical analysis and of excluding from our calcula- 
tions those that do not. Until these are solved with moderate 
completeness, our electrostatics will be applicable only ‘in gen- 
eral.’ That is, it will supply only dubious information. 

A famous crux of mathematical physics shows just how 

deceptive intuition unrestrained by reason can be. As this 

Dirichlet’s principle — was of the first importance in the evolu- 
tion of analysis in the nineteenth century, we shall describe it 
in some detail. 

By a semiphysical argument, Gauss in 1840, and W. Thom- 
son (Lord Kelvin, 1824—1907, Scotch) in 1847, using the cal- 
culus of variations, believed they had established the existence 
of a continuous solution V of Laplace’s equation having assigned 
values on any given closed surface and minimizing the integral 11 


the integration extending throughout the volume enclosed by the 
surface. It is intuitively evident from the physical situation 
of which this is the mathematical abstraction that the required 
V exists. Following Riemann (1851), we therefore assert that 
the mathematical existence of V is assured by that of the physical 
problem, and call this Dirichlet’s principle, although Dirichlet 
himself was not so rash as to state it. Dirichlet did, however, 
follow (1856) Gauss and Thomson in assuming the existence of a 
minimizing V , a much milder assumption than Riemann’s that, 
because a problem seems to make sensible physics, it must have 
a mathematical solution. 

Unfortunately for intuition, the principle in either form is 
false. Weierstrass in 1870 proved that the required minimum 
value of V is not attainable within the domain of continuous 
functions. What seemed intuitively to be a meaningful problem 
was thus shown to be a disguised incompatibility. The like 
holds for the corresponding principle in two dimensions instead 
of three. _ 

The -principle being fallacious, what is called Dirichlet s 
problem supplanted it: to find a function F(x, y, z) which, to- 
gether with its first and second partial derivatives with respect 
to x, y, z, shall be uniform (single valued) and continuous 


throughout a given closed region R, and •which shall take pre- 
assigned values on the boundary of R. 

Dirichlct’s principle was responsible for a vast amount of 
pure mathematics after Riemann’s appeal to it in the two- 
dimensional ease in his theory (1851) of functions of a complex 
variable (to be described in a later chapter). As this theory 
was one of the most extensively cultivated fields of analysis 
in the latter half of the nineteenth century, it became important 
to determine restrictions under which Dirichlet’s problem is 
solvable. The outcome was a large division of the modern theory 
of the potential. 

A list of the developers of this highly specialized topic 
reads like a directory of the leading analysts from Riemann 
(1826-1866) to Poincare (1854—1912), and down to the present. 
For a critical account to 1929, we must refer elsewhere 12 because, 
after all, potential theory is but one department of dozens in 
modern analysis, and we can attend here only to general move- 
ments. Our present interest in the subject is incidental: it is a 
typical example of the physical origin of much pure analysis, 
and of the necessity for more than acute physical intuition in the 
correct formulation and solution of important problems in 
applied mathematics. 

We note briefly the scientific and historical origins of the 
theory. Discussing Newtonian gravitation, Lagrange in 1773 
(and 1777) observed that the components of attraction at a 
given point in space, due to a distribution of mass-particles, 
arc obtainable as the space-derivatives of a certain function of 
the positions of the particles. Thus Lagrange invented what is 
now called the potential V for a Newtonian gravitational field 
due to a discrete distribution of mass-particles. Laplace (1782) 
showed that for a point in empty space the potential V due to 
a continuous distribution of matter satisfies V 2 F — 0; and 
S. D. Poisson (1781-1840, French) derived (1813) the cor- 
responding equation V 2 F = — 4rrp for points within the attract- 
ing mass, the density p at an interior point being given as a 
function of the coordinates. 

The next long step forward was taken in 1828 by G. Green 
(1795-IS4I, English), in his fundamental Essay or. ike applica- 
tion of mathematical analysis to the theories of electricity and 
magnetism. This contained the extremely useful result known 
as Green’s theorem for the reduction of certain volume integrals 
to surface integrals. 


It may be noted in passing that Stokes’ (actually Kelvin’s) 
companion “platitude of mathematical analysis” 13 for the 
reduction of certain surface integrals to line integrals, which 
also is of constant use in mathematical physics, made its first 
public appearance as a problem in a Cambridge examination 
paper of 1854. Whether any of the examinees solved the problem 
appears not to be known. But it seems likely that if anyone 
did turn in a solution acceptable to Stokes, he could not satisfy 
a modern examiner 14 with the same solution. Like Dirichlet’s 
problem, Stokes’ theorem, its proof, and its generalizations have 
developed into a thriving industry of modern analysis. A concise 
report of what has been done on this detail alone would occupy 
a chapter. 

Enough has been said to indicate the strictly physical origin 
of potential theory, in which Dirichlet’s problem is only an 
incident, although one of the first importance. It may be re- 
marked, however, that in the interests of historical justice La- 
place’s prolific equation should be renamed after Lagrange, who 
used the equation as early as 1760 in his work on hydrodynamics. 

The partial rehabilitation of Dirichlet’s discredited principle 
dates from 1899, when Hilbert proved that under suitable 
restrictions on the region in tvhich V is defined, on V itself, 
and on the values assumed by V at the boundary of the region, 
Dirichlet’s problem is rigorously solvable. But it is no longer 
intuitive in any sense. The unique historical importance of 
Dirichlet’s problem is that it was the first in potential theory 
to raise the question of existence. 

Double periodicity 

We pass on to the origins of one of the most extensive depart- 
ments of nineteenth century analysis, in which practical utility 
quickly gave way to purely mathematical interest. The history 
of multiple periodicity is a perfect foil to that of the extremely 
useful Bessel functions. 

The knowledge that many natural phenomena are periodic in 
time, or approximately so, is probably as old as the emergence of 
the human race from brutehood. Day and night, the recur- 
rence of the seasons, the waxing and waning of the moon, 
the physiology of the human body, and many other unescap- 
able facts of daily life must sooner or later have forced the 
existence of natural periodicity on even the most rudimentary 


Philosophical extrapolations of single periodicity preceded 
mathematical formulations by thousands of years. Long before 
Greece was civilized, the sublimely imbecilic vision of Plato’s 
Great Year, revived in Friedrich Nietzsche’s (1844-1900, 
German) insane dream of an Eternal Recurrence, had evolved 
from such banal phenomena as the periodicity of the seasons. 
Fortunately for the sanity of mankind, poetic philosophers 
have yet to hear of elliptic functions, whose double periodicity 
leads at once to a two-dimensional Time. In this infinitely 
ampler time, with its oo : eternity, history repeats itself indefi- 
nitely in the parallelograms of a skewed chessboard extending 
to Infinity in all directions. But the ratio of two sides of any 
lozenge is real if, and only if, the sides are parallel, when the 
ratio is Unity. 15 

The mathematics (as opposed to the mysticism) of periodic- 
ity originated in 1748 with Euler’s completely correct determi- 
nation of the values of the circular functions when the argument 
is increased by integer multiples of a half-period. Euler, inci- 
dentally, was the first to emancipate the circular functions 
from slaver}- to diagrams, and to consider them as numerical- 
valued functions of a numerical variable. The hyperbolic func- 
tions, with one pure-imaginary period, followed immediately as 
obvious consequences of Euler’s exponential forms of the circular 
functions. They are usually ascribed to V. Riccati 16 (1707-1775, 
Italian), about 1757; their simple theory was developed in 
detail by J. H. Lambert (1728-1777, German). 

None of this indispensable work suggested that more general 
functions having two distinct periods, and including both the 
circular and the hyperbolic functions as degenerate cases, might 
exist. Abel’s discovery in 1825 of these doubly periodic, or 
elliptic, functions, as they are called, is one of the outstanding 
landmarks in the history of analysis. The elliptic functions are 
of the first importance historically, not so much on their own 
account as for what they instigated. Their singularly rich and 
symmetrical theory became an invaluable testing ground for 
the vastly more inclusive theory of functions of a complex 
variable and for its prolific offshoot, the theory of algebraic 
functions. These will be considered in a later chapter; for the 
present we arc interested in the genesis of elliptic functions. 

The unfortunate term ‘elliptic integral,’ for historical 

reasons only, designates any integral of the form 


in which R(z) is a polynomial of the third or fourth degree in z 
and F(z) is rational in z. The rectification of the arc of an 
ellipse leads to a special integral of this type; hence the name. 
Of mechanical problems leading to elliptic integrals, the most 
elementary is that of finding the duration of one complete 
oscillation of a simple pendulum. 

The early work on elliptic integrals has long been of only 
antiquarian interest. A small sample will suffice to indicate 
its quality. Being unable to evaluate a special elliptic integral 
appearing in a problem of elasticity (noted in a later chapter), 
James Bernoulli in 1694 expressed his conviction that the inte- 
gration was impossible by means of elementary functions. He 
was right; but a proof of the impossibility lay far beyond his 
resources. Maclaurin (1724) translated Bernoulli’s problem into 
a geometrical construction, which would have been an advance 
had he shown what means were necessary and sufficient to 
carry it out. 

The first work to transcend the obvious was that of the 

Conti di Fagnano (1682-1766, Italian), who in 1716 proved 
that two arcs of any given ellipse may be determined in an 
infinity of ways so that their difference is a segment of a straight 
line. The significance of this is that Fagnano’s methods are 
suggestive of those used by Euler 17 in his proof (1761) of the 
addition theorem for elliptic integrals. But Fagnano’s most 
remarkable achievement was his discovery that a quadrant of a 
lemniscate can be divided into n equal parts by a Euclidean con- 
struction, where « is an integer of the form 2 m h, h = 2, 3, 5. 

As thus stated, the last may give Fagnano slightly more than 
his due; but he had the substance of it. The next published hint 
of a general theory behind such constructions occurs in the 
Disquisitions s arithmeticae (1801, p. 593, Art. 335), where Gauss 

remarks that his theory of cyclotomy “can be applied to many 
other transcendental functions [beyond the circular], for example 

to those which depend on the integral J = ==•” This 

particular elliptic integral was one of those discussed by Fagnano 
in the work just cited; its inversion leads to the special case 
of elliptic functions sometimes called lemniscatic functions. 
It would be interesting to know whether Gauss was inspired by 
Fagnano’s work; Euler frequently expressed his admiration for 
what his most sagacious predecessor in elliptic integrals had 



Another early hint of greater things to come appeared 15 in 
1771, in the discovery with which J. Landcn (1719-1790, 
English) succeeded in astonishing himself: “Thus beyond my 
expectation, I find that the hyperbola may in general be rectified 
by means of two ellipses.” Landen's ingenious analytic reformu- 
lation (1775) of his geometrical theorem is recast today in the 
transformation of the second order (more generally, of order 
2 n , n an integer) in elliptic functions. 

But all of this early work, including much by d’Alembert, 
was haphazard in comparison with Euler’s systematic attack 
on elliptic integrals and their geometrical applications. Em- 
bedded in an enormous mass of hideous formulas and intricate 
calculations, two items in Euler’s contribution outrank all the 
rest in historical significance. The first "was the addition theorem 
(1761) for elliptic integrals, rated by Euler’s contemporaries 
and immediate successors as the most amazing tour de force 
of manipulative skill in eighteenth-century analysis. 

Euler’s second major contribution was of far greater impor- 
tance both historically and mathematically, as by an almost 
ludicrous mischance of fate it misdirected progress for all of 
forty years after his death in 1783. In the introduction to a 
memoir 19 of 1764, Euler advocated the incorporation of elliptic 
arcs into analysis on a parity with logarithms and circular arcs. 
(Note the italicized word.) Abandoning the fruitless efforts of 
his predecessors and contemporaries to integrate elliptic differ- 
entials in finite terms by means of functions then known, Euler 
boldly proposed that elliptic integrals be recognized as primitive 
new transcendents to be investigated on their own merits. 
If this is not what he meant, he proceeded in all of his own 
analysis as if it were. So great was the momentum of Euler’s 
algoristic ingenuity that before he could realize his initial 
mistake he was carried completely out of sight of the right 
turning which he had missed. That he, of all mathematicians, 
should have gone astray in this particular matter is one of those 
mysteries in the evolution of mathematics that pass all under- 
standing. The master who had initiated the modern theory of the 
circular junctions failed to observe the greater opportunity 
which his Providence kept crowding on him and which, had he 
given it even a casual glance, must have appeared to a mathe- 
matician of his particular quality as the most natural thing in 
the world. Instead of considering elliptic arcs as the basic new 
transcendents, and thereby endowing an already overburdened 


integral calculus with a new wealth of uncouth formulas, Euler 
might easily have followed the simple lead of trigonometry. His 
oversight in adopting the elliptic integrals instead of their 
corresponding inverse functions as the data of his problem led 
him into a morass of tangled algebra, precisely as if he had at- 
tempted to develop trigonometry by an exclusive use of the 
inverse circular functions — his ‘circular arcs’ — sin -1 x , cos -1 x 
tan -1 x, etc. The far greater complexity of the theory of elliptic 
over circular arcs bogged him deeper at each step. 

Realizing that Euler’s heroic explorations in the wilderness 
of elliptic integrals had not got the undauntable pioneer very 
far in spite of many treasures found along the way, A. M. 
Legendre (1752-1833, French), in 1786 set out on his own 
explorations. For nearly forty years he followed Euler’s 
trail, systematizing and civilizing as he went. It is at least 
conceivable that uncritical reverence for the works of his great 
predecessor was partly responsible for Legendre’s personal 

More systematic than Euler, and taking more time to his 
work, Legendre reduced his chaos of refractory material to as 
coherent a whole as seems to be possible. To him are due the 
three standard forms of elliptic integrals to which any elliptic 
integral is reducible. Legendre’s integrals are of course not the 
only canonical forms possible, and many others have been pro- 
posed; but Legendre’s retain their usefulness. Forty years 
of unremitting labor by a master could not fail to produce 
much of value, if only for its suggestiveness. In particular, 
Legendre’s work on the algebraic transformation of elliptic 
integrals directly inspired Jacobi’s first notable success. 

Legendre presented systematic accounts of his theory in 
1811-17, in his Exercices de calcul integral sur divers ordres de 
transcendentes et sur les quadratures, amplified in 1825-32 in the 
three volumes (with supplements), Traite des fonctions elliptiques 
ct des inlegrales euleriennes. The title of the second is responsible 
for a prevalent confusion in some historical accounts: Legendre s 
personal work is concerned with elliptic integrals, not with 
elliptic functions. The distinction, which became of epochal sig- 
nificance in 1827 with the publication of Abel’s inversion of 
elliptic integrals, is comparable to that between night and day. 
Before Abel, nothing was publicly known of elliptic functions 
as they did not exist outside the private papers of Gauss unti 
Abel invented them. 


In addition to providing invaluable hints to Abel and Jacobi 
for the theory of elliptic functions, Legendre’s treatises furnished 
Cauchy and others with numerous definite integrals, explicitly 
evaluated, on which to test the efficiency of integration by 
Cauchy’s method of residues. The like holds for Legendre’s 
systematization of the beta and gamma functions as they 
existed in his day. But here again it must be remembered that 
Legendre’s analysis of 1827 became hopelessly archaic with the 
creation of modern methods by the great analysts of the nine- 
teenth century, beginning with Cauchy in 1825. The contrast 
between the old and the new is strikingly evident on comparing 
Legendre’s discussion of the gamma function with that of 
Weicrstrass in 1856, only twenty-three years after Legendre’s 

In taking leave of this fine mathematician of the eighteenth 
century, we may remember him as a man of the highest char- 
acter, whose only ambition was the advancement of mathe- 
matics. If Legendre was so far outdistanced in his own lifetime 
by younger men — Gauss in arithmetic and the method of least 
squares, Abel and Jacobi in elliptic functions — it was partly 
because his own labors had laid the necessary steppingstones. 
And although Legendre misjudged Gauss and hated him with a 
venomous hatred, he was the first to welcome and publicize the 
works of Abel and Jacobi which rendered obsolete his own 
efforts of forty years. The veteran of seventy-odd not only 
showed himself incapable of jealousy for his vigorous young 
rivals in their early twenties, but took pains to understand their 
work and to expound it in an amplified edition of his own. Such 
liberality of spirit is no commoner in mathematics than it is 

Abel revolutionized the subject, and at the same time 
opened the floodgates of nineteenth-century analysis, in 1827 
with a simple remark, “I propose to consider the inverse func- 
tions.” Instead of regarding the elliptic integral 

p dx 

“ ' J V(T-~ckri)(l -f dr) 

as the primary object of investigation in which a is considered 
as a function a(.x) of .v, Abel reversed the problem and regarded 
•v as a function, which he denoted by of a. This inversion 
of the integral was the essential first step which Abel’s pred- 


ecessors had overlooked. Its ‘naturalness’ after it had been 
taken was obvious from the analogy with (* properly restricted) 

Abel’s first capital discovery 20 concerning the new functions was 
their double periodicity: <£(# + pi) = 4>(x), 4>(x -f pi) = <j>(x) 
where pi, pi arc constants whose ratio is not a real number. Thus 
the elliptic function </>(#) is doubly periodic . 

Impressed by the great wealth of new ideas that entered 
mathematics as a direct consequence of Abel’s simple remark, 
Jacobi some years after Abel’s death characterized inversion 
as the secret of progress in mathematics: “You must always 
invert.” If science or mathematics presents us with an awkward 
situation in which y is given as a function of x, say y = /(*), 
we should examine the inverse situation, x = / -1 (y), as Abel did 
when he inverted elliptic integrals and discovered that the 
inverse functions — the elliptic — are doubly periodic. Jacobi, 
who balanced his enthusiasm for mathematics with a sense of 
the ridiculous and who kept his tongue in his cheek when he 
pontificated, did not intend his prescription to be gulped down 
as a panacea. He was one of the least professorial of professors 
who ever lectured to an advanced class. 

Jacobi’s classic Fundamenla nova theoriac functionum cllipti- 
carum , published in 1829, the year of Abel’s death, exploited 
the consequences of inversion and double periodicity, and made 
the new functions easily accessible to the mathematical public. 
Even if, as is now generally conceded, Abel’s was the priority 
in the two basic discoveries, Jacobi made the theory his own 
and contributed enough to entitle him to rank with Abel as one 
of its creators. 

In awarding priority to Abel at the expense of Gauss, we 
have followed the modern custom of dating ownership from first 
publication. But with the printing of Gauss’ posthumous papers 
and the scientific diary which he kept as a young man, it is 
known that Gauss was in possession of the double periodicity 
of the lemniscatic function in 1797. Early in 1800 he had dis- 
covered the general doubly periodic functions, anticipating 
Abel by a quarter of a century. His posthumous papers also 
contain numerous formulas relating to the elliptic theta con- 
stants, rediscovered and brilliantly applied by Jacobi. But 
Gauss, possibly for lack of an opportunity to develop an 


systematize his discoveries, published nothing on elliptic func- 
tions. Nor did he make any public claim 21 to have anticipated 
Abel and Jacobi. In estimating the place of Gauss in mathe- 
matics, it is customary to credit him with what he actually did. 
Thus non-Euclidcan geometry and elliptic functions are two 
of the items which have counted in ranking Gauss with 
Archimedes and Newton, although he published nothing on 

Elliptic functions have been given more space than their 
position relative to modern analysis might justify in a general 
account, because they clearly mark the beginning of a prolific 
epoch and were responsible for several major activities in the 
algebra, arithmetic, and analysis of the nineteenth century. 
Double periodicity not only opened up boundless new terri- 
tories; it also marked the definite end of a road which had been 
followed since Euler’s creation of analytic trigonometry. Jacobi 
proved (1834) that, if a single-valued function of one variable 
is doubly periodic, the ratio of the periods cannot be a real 
number; and that single-valued functions of one variable having 
more than two periods are impossible. 22 

Further technicalities would take us too far off the main 
road. However, three details of Abel’s and Jacobi’s early work 
were to prove so prolific of new mathematics all through the 
nineteenth century that they must be mentioned in passing. 
The first is Abel’s discovery of complex multiplication, most 
conveniently described in terms of the Wcierstrassian elliptic 
function p(«) 5 Esp(ttj«i, w : ), with periods 2«i, 2a?j, arising from 
the inversion of a certain standard elliptic integral involving 
the square root of a polynomial of degree three. The choice of 
p(») implies no restriction. If n is a rational integer, p(nu) is 
expressible as a rational function of p («). Seeking all other n’s 
for which a similar theorem holds, Abel found the following 
unexpected 25 result. If c is a complex number such that p(ca|«i, 
ut) is rationally expressible in terms of p(«Jo>j, w : ), p(«) is said 
to admit a complex multiplication by c. In order that such a c 
may exist, it is necessary and sufficient that Wi/oj; be a root of 
an irreducible algebraic equation of the second degree with 
rational integer coefficients. This should be enough to suggest 
that the theory of complex multiplication is intimately connected 
with the arithmetic of binary quadratic forms. 24 The develop- 
ment of this hint occupied scores of algebraists, beginning with 
Kroncckcr (1857) and Iicrmitc (1859). 


The second item is Jacobi’s representation of his doubly 
periodic functions as quotients of what are now called elliptic 
theta functions. 25 The thetas are not doubly periodic; one of 

Jacobi’s four is # 3 (*|r) = ^ q ni cos 2mrx, where q ~ e irt 

r s co 2 /coi, \q\ < 1. The others are obtainable from this by 
simple linear transformations on x , for example, 

&s(x + ijr) s t? 4 (*|t). 

As the values of x for which the thetas vanish are readily deter- 
mined, the analytic character 25 of the elliptic functions is put 
in evidence, and from this the Fourier expansions are obtained. 
The corresponding theta constants ( x — 0) had been investi- 
gated by Euler in the 1750’s and by Gauss 27 about 1800. But it 
was only when Jacobi discovered their connection with elliptic 
functions that their symmetrical theory emerged. Apart from 
their own extensive theory, the elliptic theta functions proved 
of great importance as clues to more general theta functions. 
These will be noted when we come to functions of a complex 

The third advance that opened up another vast expanse of 
nineteenth-century analysis also originated with Abel and 
Jacobi. It is required to exhibit the connections (algebraic 
relations) between elliptic functions, or between theta functions, 
whose respective pairs of periods are obtained from one another 
by linear homogeneous transformations with rational integer 
coefficients and non-vanishing determinant. This, the trans- 
formation theory, includes as special cases the problem of real 
multiplication, as for p(nu ) noted above, and that of division of 
the periods by a rational integer. As will appear later, a single 
detail of this theory, that of the elliptic modular functions, 
expanded in the late nineteenth century into an independent 
branch of mathematics. Its connection with the general quintic 
was noted in an earlier chapter. As might be anticipated from 
its formulation, the general problem draws heavily for its 
modern solution on the theory of linear groups. 

Even these meager hints, displayed against a background of 
unprecedented activity in all departments of mathematics, 
should suffice to suggest the extent and intricacy of the theories 
that evolved from Abel’s discovery of double periodicity. 
Each of half a dozen or more leaders elaborated the entire 
theory 28 or some favored subdivision according to his persona 


conception of symmetry and grace. Eighty years of this rugged 
aestheticism endowed analysis with a welter of conflicting 
notations and trivial distinctions without much of a difference, 
through which even an expert picks his way with exasperation. 
Almost in spite of themselves, the leaders rapidly acquired hosts 
of partisan followers. Mathematicians of all capacities began 
swarming into the new territory within a decade of its discovery. 
Several were pupils of Jacobi, but others quickly found leaders 
with different ideals. 

The causes of this mass migration are not far to seek. 
Unlike the special functions devised primarily for the solution 
of physical problems, such as the Bessel coefficients, the elliptic 
functions seemed to have been created to round out and extend 
the integral calculus as it had evolved since the days of Newton 
and Leibniz. With the applications to the theory’ of numbers by' 
Kronccker and Hermitc in the late 1850’s, it seemed also as if 
Gauss must have elaborated his arithmetical theory of binary 
quadratic forms especially’ for these unforeseen consequences of 
Abel’s and Jacobi’s early’ discoveries. Comprehensive syntheses 
to correlate these unexpected coincidences were sought as the 
century' aged, and were found in the Galois theory of equations, 
the algebraic theory of fields, and the arithmetic of quadratic 
number fields. 

Algebraic curves and surfaces 15 also absorbed enormous 
quantities of elliptic functions. Applications to classical applied 
mathematics were made simultaneously, especially to rigid 
dynamics and problems in potential theory'. But it must be 
admitted that most of these practical applications have remained 
of greater interest to pure mathematicians than to working 
scientists. The rotation of a rigid body', 30 for example, yields 
numerous elegant exercises in the elliptic theta functions; but 
few engineers who must busy’ themselves with rotation have time 
for elegant analysis. The like holds for the occurrence of elliptic 
functions in practical applications of conformal mapping. When 
faced with one of these enticing horrors, the experienced designer 
turns to his drafting board. Contrasted with the Bessel functions, 
the elliptic functions arc incomparably’ more beautiful and less 
useful. Yet — or possibly' on that account — they were preferred 
many’ to one by’ the leading mathematicians of the nineteenth 
century because, in a sense that any mathematician will un- 
derstand, they were closer to the ‘natural’ development of 


Differential and Difference 

Continuing with the mathematics directly inspired by 
science, we shall indicate next four of the principal stages, 
not discussed in other connections, by which differential equa- 
tions became a major discipline of modern pure mathematics. 
In its later development, this great episode is complementary 
to the evolution of the Galois theory of algebraic equations 
and the emergence of algebraic structure. Once more we shall 
see mere ingenuity being gradually displaced by coordinated 
attacks, and again we shall note the distinction between mathe- 
matics as practiced in the recent period and nearly all that 
preceded the nineteenth century. 

Discounters of ingenuity do not mean to disparage intuition 
and insight in any assault on basically new problems. They 
merely emphasize that the characteristic strategy of modern 
mathematics favors the mass attack, where feasible, rather than 
any number of brilliantly executed raids. General methods, not 
individual gains, are the order of the modern day in mathe- 
matics. Ingenuity still has its function, even in a general offen- 
sive; but it is of a more comprehensive kind than any that 
sufficed in the past. The problems of modern mathematics are 
not isolated, and to overcome them coordinated efforts on a 
wide front are increasingly necessary. 

Five stages 

The first stage in differential equations, opening with 
Leibniz in the 1690’s, closed about seventy years later. Roughly, 
what was accomplished in this period amounts to the first eig 



weeks’ work in the usual introductory college course. Remember- 
ing that mechanics, dynamical astronomy, and mathematical 
physics were intensively cultivated all through and after this 
period and that numerous problems of analysis originated thus, 
we must also bear in mind that there was no adequate discussion 
of differential equations before Cauchy in the 1820’s obtained 
the first existence theorems. This inaugurated the second stage. 
The third opened in the 1870’s-80’s with the application by 
M. S. Lie (1842-1899, Norwegian) of his theory of continuous 
groups to differential equations, particularly those of Hamilton- 
Jacobi dynamics. The fourth stage, beginning in the 1880’s 
with the work of E. Picard (1856-1941, French) developed natu- 
rally from the third. Here the aim was to construct for linear 
differential equations an analogue of the Galois theory of 
algebraic equations. 

Each stage after the first marked a definite and abrupt 
advance. The second paralleled the rigorizing of the calculus 
by Cauchy, and might have been anticipated from the general 
trend in analysis. The third, Lie’s, even in retrospect, appears to 
have been unpredictable. Each of the periods has left a sub- 
stantial residue in living mathematics; the last three posed 
many problems which still engage scores of specialists. What 
may be the beginning of a fifth stage opened in the 1930’s, 
paralleling the modern development of abstract algebra. We 
pass on to a brief indication of some of the outstanding acquisi- 
tions in each of these stages and the accompanying developments 
in finite differences. Before the last three stages can be dis- 
cussed, the concept of invariance must be described. This will 
be done in the next chapter. 

The reign of formalism 

Both Newton and Leibniz in the seventeenth century solved 
simple ordinary differential equations of the first order. It 
seems to have been believed in this earliest stage that the func- 
tions then known would suffice for the solution of the differential 
equations arising from problems of geometry and mechanics; 
and the aim was to find such explicit solutions, or to reduce the 
solution to a finite number of quadratures. Even when a solution 
was exhibited as a quadrature, it does not seem to have been 
suspected that the required integration might necessitate the 
invention of new transcendents. In fact, it was not until the 
ISSO's — a stretch of two centuries — that definite knowledge 


concerning the extreme rarity of differential equations integrate 
in this rudimentary sense was obtained. Very roughly, if a 
differential equation is written down at random, the odds against 
its being solvable in terms of known functions or their integrals 
are infinite. 

The first faint hint of generality was Newton’s (1671) clas- 
sification of ordinary differential equations of the first order 
into three types, and his method of solution by infinite series. 1 
The coefficients of the assumed power-series solution were found 
as usual. There was no discussion of convergence. Without 
explicit statement of the assumption, it was assumed that the 
existence of a physical problem guarantees the existence of a 
solution of the equivalent differential equation. This seemingly 
reasonable supposition remained unquestioned in applied mathe- 
matics from Newton to Riemann. Its viciousness was first 
unmasked, as we saw in connection with Dirichlet’s principle, 
only in 1870. 

Another early forward step was taken by Leibniz, who 
stumbled on the technique of separating variables. Nearly two 
centuries were to elapse before Lie’s theory showed when and 
why this familiar device should succeed. Among other early 
advances, the homogeneous linear differential equation of the 
first order was reduced to quadratures by Leibniz (1692); and 
James Bernoulli (1690) solved the equation of the tautochrone 
by separation of variables. His brother John (1694) circum- 
vented dx/x, not well understood at the time, by first applying 
an integrating factor. Incidentally, the discovery of integrating 
factors proved almost as troublesome as solving an equation. 
Another hint of more general tactics appeared in Leibniz’ (1696) 
change of the dependent variable. John Bernoulli also used this 
device. By the end of the seventeenth century all the usual 
elementary and inadequate tricks for first-order equations were 

In addition to problems of the differential calculus on tan- 
gents, normals, and curvature of the types common as exercises 
in textbooks, the calculus of variations also had stimulated in- 
genuity in solving differential equations. Thus James Bernoulli s 
(1696) isoperimetric problem (noted in another connection) 
led to a differential equation of the third order which John 
reduced to one of the second. 

Before 1700, John also attacked the general linear homo- 
geneous differential equation with constant coefficients, lo 


dispose of this detail here, a complete discussion of such equa- 
tions was given in 1743 by Euler, who also (1741) devised the 
classical method for non-homogeneous linear equations. 

The name of Count Riccatr (1676—1754, Italian) is familiar 
to every student in a first course. What is usually called Riccati’s 
equation (1723) persistently defied solution in finite form. In 
accordance with the taste of the age, the ‘real’ problem was to 
impose sufficient restrictions on the variables to render the 
transformed equation finitely solvable by separation of varia- 
bles. The Bcrnoullis claimed to have at least partial solutions; 
and in 1725 Daniel noted that if m is of the form —4n/(2n ± 1), 
where it is a positive integer, dyjdx -f- ay- — bx n is solvable in 
finite terms. By 1723 at latest, then, it was recognized that even 
an ordinary differential equation of the first order docs not 
necessarily have a solution finitely expressible in terms of 
elementary functions. But anything approaching a proof of the 
impossibility, in general, of such a solution lay far in the future. 

Singular solutions were noted unexpectedly early, the first 3 
instance being due to Taylor (of Taylor’s series) in 1715. Clair- 
aut, whose name decorates a special type of equation in a first 
course, followed in 1734 with